Gravitational Instabilities
in Circumstellar Disks
Kaitlin M. Kratter1 & Giuseppe Lodato2
arXiv:1603.01280v1 [astro-ph.SR] 3 Mar 2016
1
Department of Astronomy and Steward Observatory, Univ. of Arizona, 933 N
Cherry Ave, Tucson, AZ, 85721
2
Dipartimento di Fisica, Università degli Studi di Milan, Via Celoria 16, 20133
Milan, Italy
Xxxx. Xxx. Xxx. Xxx. YYYY. 00:1–41
Keywords
This article’s doi:
10.1146/((please add article doi))
star and planet formation, accretion disks, hydrodynamics
c YYYY by Annual Reviews.
Copyright All rights reserved
Abstract
Star and planet formation are the complex outcomes of gravitational
collapse and angular momentum transport mediated by protostellar
and protoplanetary disks. In this review we focus on the role of gravitational instability in this process. We begin with a brief overview of
the observational evidence for massive disks that might be subject to
gravitational instability, and then highlight the diverse ways in which
the instability manifests itself in protostellar and protoplanetary disks:
the generation of spiral arms, small scale turbulence-like density fluctuations, and fragmentation of the disk itself. We present the analytic
theory that describes the linear growth phase of the instability, supplemented with a survey of numerical simulations that aim to capture the
non-linear evolution. We emphasize the role of thermodynamics and
large scale infall in controlling the outcome of the instability. Despite
apparent controversies in the literature, we show a remarkable level of
agreement between analytic predictions and numerical results. In the
next part of our review, we focus on the astrophysical consequences of
the instability. We show that the disks most likely to be gravitationally
unstable are young and relatively massive compared to their host star,
Md /M∗ ≥ 0.1. They will develop quasi-stable spiral arms that process infall from the background cloud. While instability is less likely
at later times, once infall becomes less important, the manifestations
of the instability are more varied. In this regime, the disk thermodynamics, often regulated by stellar irradiation, dictates the development
and evolution of the instability. In some cases the instability may lead
to fragmentation into bound companions. These companions are more
likely to be brown dwarfs or stars than planetary mass objects. Finally,
we highlight open questions related to (1) the development of a turbulent cascade in thin disks, and (2) the role of mode-mode coupling in
setting the maximum angular momentum transport rate in thick disks.
1
We dedicate this article to the memory of Francesco Palla, who has been a source of inspiration
for star formation and disk studies throughout the years. Through his support and encouragement
of young scientists, his contributions live on.
1. INTRODUCTION
Protostellar and protoplanetary disks stand at the center of our study of origins. They regulate
the formation of stars and the planetary systems that they host. Over the last three decades,
the observational study of protostellar disks has evolved from quantifying hints of infrared excess
(Kenyon & Hartmann 1987) to resolving ring-like structures at 1AU scales (ALMA Partnership
et al. 2015). Long wavelength studies with facilities such as the SMA, Plateau de Bure, CARMA,
and now ALMA, combined with sensitive infrared measurements from IRAS and Spitzer enable
detailed studies of the sizes, temperatures, and masses of a variety of disks in young clusters. While
many uncertainties remain in translating high resolution interferometric measurements into reliable
disk properties, we can begin to see the various stages of star and planet formation occurring in
these disks.
Theoretical studies have made similar progress, from simple, 1-D diffusive models to threedimensional (magneto)hydrodynamics simulations that incorporate radiative transfer. These advances have been enabled by the burgeoning computational resources at our disposal. While the
most complex of these models provide many insights, which we review in detail, simple models
prove valuable to this day.
In this review we discuss the role that self-gravity plays in regulating the growth and evolution
of protostellar and protoplanetary disks. Self-gravity in general refers to the mutual gravitational
influence of relatively diffuse gas on itself, in contrast to the gravity exerted on the gas by an
external body. In this context, self-gravity refers to the effect of the gravitational field of the
disk itself, rather than the protostar, on the evolution of the system. Although the dividing line
between protostellar and protoplanetary disks is fuzzy, we use the former to refer to younger disks
surrounding protostars that are not yet near their final mass, while the latter refers to the remnant
disk in which the planet formation process completes (of course it may begin much earlier). The
physics we discuss is relevant for all stellar masses, but much of our scalings are for sun like stars.
We focus our attention on hydrodynamics, and thus we do not discuss the important role that
magnetic fields and MHD effects have on the evolution of protostellar disks. It is well known that the
hotter parts of protostellar disks are subject to the magneto-rotational instability (Balbus & Hawley
1998), that can provide the bulk of angular momentum transport. On the other hand, the colder,
outer disks, which are the regions most likely to become gravitationally unstable, are probably less
affected by MHD turbulence, given their low ionization state. Very few numerical studies have
considered the combined effects of MHD and gravitational instabilities in disks (Fromang et al.
2004a,b), whose interplay might be important in protostellar disk evolution.
In Section 2 we begin with a brief observational overview with an eye towards the applicability
of the physics of self-gravitating fluids to observed systems. In Section 3 we provide an in-depth
review of the nature of self-gravity and gravitational instabilities, including the non-linear outcome
of the instability: fragmentation of a disk. The physics discussed here is applicable to accretion
disks in many different astrophysical contexts. In Section 4 we hone in on disks around young stars,
and consider how they fit into the various regimes outlined in Section 3. In Section 5 we discuss
the relevance of gravitational instabilities to the formation of planets both directly (gas-collapse)
and indirectly (concentration of dust particles). We conclude with a brief summary and prospects
for future.
2. Observational Evidence for Self-Gravitating Disks
Before reviewing the theoretical underpinnings of the role of self-gravity in protostellar and protoplanetary disk evolution, we address the observational evidence that this physics is relevant. As we
detail in later sections, understanding the behavior of self-gravitating disks is complex and depends
www.annualreviews.org • GI DISKS
3
on a range of disk properties including size, surface density, temperature, and thermal physics. The
standard reference for quantifying the degree to which a disk is self-gravitating is the parameter Q,
defined as (Toomre 1964):
cs κ
(1)
Q=
πGΣ
where cs is the sounds speed, κ is the epicyclic frequency (which is equal to Ω in a Keplerian disk),
and Σ is the disk surface density. As Q → 1 self-gravity becomes increasingly important. We
discuss the origin of Q in more detail in section 3. To translate this constraint into one motivated
by observations, we note that Q can be written as:
Q=
M∗ H
cs Ω
=f
πGΣ
Md r
(2)
where M∗ is the stellar mass, Md is the disk mass, H is the disk scaleheight, and r the radius.
We subsume numerical factors of order unity, which depend on the surface density profile, into a
pre-factor f . While we cannot escape the dependence on the disk aspect ratio, H/r, which depends
on the temperature, we can use this relation to obtain an order of magnitude estimate for the
minimum mass required for a disk to be self-gravitating. Colder, thinner disks are more prone
to self gravity. Star forming regions typically do not have temperatures below 10K, so we take
that as a lower limit. What about disk radii? Many lines of argument, from measures of cloud
angular momentum (Goodman et al. 1993), to stellar interactions (Adams et al. 2006), to direct
observations in both absorption (Eisner et al. 2008) and millimeter wavelengths (Andrews et al.
2011) suggest that 102 − 103 AU are reasonable limits for disk radii. At constant temperature, the
aspect ratio increases with disk radius as r1/2 , so we use 100AU to set H for our scaling. The
requirement that Q < 1 (implying gravitational instability, see below) then implies:
Md
> 0.06
M∗
1/2 1/2
f
T
r 1/2 M
1
10K
100AU
M∗
(3)
Thus for self-gravity to be important, i.e. for Q to be near unity given optimistic constraints on
temperature, we expect disk-star mass ratios greater than ≈ 10−2 .
With this benchmark in mind, we now consider the evidence for disks with mass ratios in this
regime or higher.
2.1. The Phases of Disks
Early studies of protostellar disks in the infrared engendered a classification system based on the
slopes of disk+star spectral energy distributions (SEDs). Three classes, 0, I, and II, correspond to
positive, flat, and negative power law indices longward of 2 microns. In addition there is a final
phase, Class III, wherein the SED is dominated by the stellar photosphere, but weak Infrared excess
indicates remaining disk material. There is some evidence that this sequence corresponds with
evolutionary states. Class 0 protostellar sources correspond to young, deeply embedded objects,
whose disk-like component is hard to measure. Class I sources are thought to have similar mass in
the envelope and disk, and Class II sources have mostly finished accreting from their natal envelope,
revealing just the protoplanetary disk, as most of the stellar accretion has been completed. Class
III disks are not thought to be self-gravitating.
2.2. Measuring Disk Masses
Measuring disk masses at any evolutionary state remains extremely challenging. Even as observatories like ALMA grow ever more sensitive, several issues limit precise mass measurements. Many
4
Kratter & Lodato
limitations are related to the fact that most disk masses are measured by proxy through detection
of millimeter wavelength fluxes associated with dust grains of up to a millimeter in size. Based on
studies of the ISM, the ratio of gas to dust is thought to be of order 102 , but this is not well confirmed
in disks. Secondly, because larger grains and rocks do not emit at millimeter wavelengths, these
data do not reveal the potentially sizeable fraction of dust which has already grown, something for
which there is already strong evidence (e.g., Pérez et al. 2012). Moreover, the conversion of a flux
to a <mm-size dust mass requires adopting an opacity, which is itself dependent on unknowns like
the grain-size distribution, and grain properties such as chemical composition and porosity (see, e.
g. Natta et al. 2004; Ricci et al. 2010). Finally, mass measurements from dust are only possible
if the dust is optically thin. If the dust is optically thick, which is possible in the inner regions of
disks (soon to be resolved by ALMA) only a lower limit on the mass is possible.
In addition to the uncertainties generated by the dust properties, Dunham et al. (2014) have
highlighted the estimation errors introduced by contamination of disk flux measurements due to
the presence of a core at early phases, leading to overestimated masses at early times. Similarly,
as disks grow to larger radii, high resolution interferometric studies can also resolve out large scale
structures where much of the mass resides. This error leads to a more severe underestimate of disk
masses.
With these caveats in mind, we review the best estimates of disk masses.
2.2.1. Observational Results. The most comprehensive survey for disk masses at this time is that
of Andrews et al. (2013), who survey all known members of the nearby star forming region Taurus.
These disks are mostly Class II sources, however some class I sources are included.
Andrews et al. (2013) compiles photometry at 1.3mm using the SMA to derive both disk masses,
and disk-star-mass ratios. The stellar masses are estimated using the isochrones from D’Antona &
Mazzitelli (1997); Baraffe et al. (2002) and Siess et al. (2000). Disk masses are estimated from the
submillimeter flux assuming a constant distance of 140pc, dust-gas ratio ζ = 0.01, dust opacity:
κν = 2.3cm2 /g and a disk temperature constant with radius, which scales with stellar luminosity
as hTd i = 25(L∗ /L )1/4 K. The masses are derived from the following equation for optically thin,
isothermal dust:
log Md = log Fν + 2 log d − log(ζ · κν ) − log Bν (hTd i),
(4)
where Fν is the observed flux, Bν is the Planck function and d is the distance to the source.
It is apparent from Andrews et al. (2013) figure 11 that most disk masses are far too low in mass
by 2-3 Myr (predominantly class II stage) for GI to be relevant. Note additionally that the assumed
outer disk temperatures for this sample are higher than our optimistic (for triggering GI) estimate
of 10K. Depending on order unity factors, at most 20%, and probably more like 10% of Class II
disks are plausibly massive enough for self-gravity to be important. A self-consistent treatment of
the disk temperatures in our instability threshold (Equation (3)) and mass estimates would yield
critical mass ratios closer to 0.1, higher than nearly every disk in the sample.
As this sample is dominated by Class II disks, we may conclude that both angular momentum
transport and planet formation reliant on strong self-gravity are rare at best for systems which have
evolved to the Class II stage.
What about evidence for younger, massive Class I and Class 0 disks? Unfortunately at the
time of this writing the evidence remains sparse. Class 0 and I sources are deeply embedded and
thus intrinsically harder to both identify and measure. Eisner (2012) and Sheehan & Eisner (2014)
have isolated and modelled in detail the Class I sources in Taurus. They find that the masses are
typically higher than in the Class II phase by a factor of two: the median mass of Class I disks is
0.008M compared to 0.005M for older sources. Among Class I sources, roughly 10% have masses
higher than 0.01M (Eisner 2012), at the margins of susceptibility to strong self-gravitating effects.
www.annualreviews.org • GI DISKS
5
Some individual sources have been measured to have relatively massive and compact disks in the
Class 0/I phase, for example IRAS 16293-2422B (Rodrı́guez et al. 2005), or WL12 (Miotello et al.
2014).
A recent survey by Mann et al. (2015) also hints that younger sources may indeed have more
massive disks. These authors find that in NGC2024 (thought to be < 1Myr old as compared to
2-3 Myr for Taurus) as many as 10% of disks have masses greater than 0.1M , with 20% above
0.01M . Despite the 2σ significance of higher mass measurements, selection biases alone (related
to stellar mass and multiplicity) could account for the increase if the cluster properties of Taurus
and NGC2024 are similar.
Finally, we turn to observations of disks in the Class 0 stage. The most conclusive evidence
for a relatively massive disk comes from Tobin et al. (2013). Using the SMA and CARMA, they
detect an edge on ≈ 150 − 200AU disk surrounding the Class 0 protostar L1527. Using a similar
optically thin dust model they derive a mass comparable to Class I sources of 0.075M , however
the protostellar mass is thought to be < 0.5M , and thus the disk-star mass ratio is easily high
enough for self-gravity to be important. There are several other tentative detections of Class 0
disks (see e.g. Looney et al. 2000; Enoch et al. 2009). A larger survey in Perseus by Tobin et al.
(2015) resolved several more likely candidates and found that more than 50% of 9 sources were
more consistent with envelope + disk models rather than pure envelope models. Additionally
several sources show evidence for rotation (though not necessarily Keplerian motion). The masses
measured in this survey cannot reliably separate out all envelope contamination, thus the increase
in masses (up to 0.1M ) is not necessarily indicative of strong self-gravity. Nevertheless it is quite
plausible that disk masses are higher in the Class 0 phase.
Maury et al. (2010) have made strong claims against the presence of disks in the Class 0 phase,
but the well-resolved L1527 was included in their sample, suggesting that perhaps the data was
insufficient to rule out some extended disks. Tobin et al. (2013) had slightly better resolution, and
also a different beam alignment relative to the disk direction, which might explain the Maury et al.
(2010) non-detection.
While dust mass measurements are most common, recent efforts have been made to directly
detect molecular gas tracers to constrain the total mass and dust-to-gas ratio. Bergin et al. (2013)
use Herschel data to detect HD in TW Hya, finding a relatively large disk mass for a nominally 10
Myr system. Williams & Best (2014) used optically thick and thin CO isotopologues and conclude
that most Class II disks have dust-to-gas ratios up to a factor of 10 below ISM values, implying
that either planet formation is well underway, or total disk masses are even smaller.
The above discussion focused on solar type and Herbig AeBe stars, for which there is more
data, but there is some evidence that high mass stars host massive, self-gravitating disks (Cesaroni
et al. 2007 and references therein). Precise measurements remain elusive because high mass stars
are typically further away, and thus harder to resolve. Additionally, the only evidence for disks
coincides with the equivalent of the Class 0-I phases, when the stars are deeply embedded in a gas
envelope. For example, rotationally supported disks are difficult to distinguish from rotationally
flattened envelopes (Johnston et al. 2011). One of the more convincing examples (Shepherd et al.
2001, 2004) reveals a rotating massive disk around a B-star that is likely gravitationally unstable
based on the estimated parameters. Krumholz et al. (2007) has shown that a fully functional ALMA
may be able to validate these observations using molecular lines.
2.3. Direct Evidence for Disk Self-Gravity
While the population at large may not show strong evidence for disk self-gravity, a handful of
individual sources do show indications of the spiral arm structure typically associated with spiral
6
Kratter & Lodato
density waves. These wave may be triggered either by a young planet, or by disk self-gravity. Muto
et al. (2012) and Wagner et al. (2015) find evidence for spiral structure in scattered light images of
transition disks. Self-gravity might be at play, but the low sub-mm masses make this questionable.
Benisty et al. (2015) also find evidence for spiral structures in scattered light images, but claim a
planetary origin. ALMA data might settle these two theories by providing disk mass constraints,
but at present neither theory can be ruled out (Dong et al. 2015).
2.4. Indirect Evidence for Disk Self-Gravity
While observational examples of disks with strong self-gravity remain scarce, there are other indirect sources of evidence from comprehensive studies of star formation. There are a variety of
reasons to expect that disks may be more massive at earlier evolutionary states. First, since the
discovery of the first statistically significant sample of protostars, observers and theorists alike have
pointed out the “luminosity problem” (Hartmann & Kenyon 1996). This problem arose due to the
mismatch between the observed protostellar luminosities, of the order of a few L , and the expected
luminosities derived from basic energetic arguments that predict that the accretion luminosity:
Lacc ≈
M
Rin
Ṁ
GM∗ Ṁ
≈ 100L
Rin
M 10−6 M /yr 4R
(5)
where Rin is the inner disk radius and Ṁ is the infall rate, estimated based on a star formation
timescale of roughly 1Myr. This over prediction of the typical luminosity has been used as evidence
for large, short-lived accretion events, either bursts, or more continuous but rapid events early in a
protostars history. A variety of theories have been proposed for producing these bursts.
Before considering specific theories, we note the following simple scaling argument. Consider a
star forming core or turbulent filament. Observational evidence suggest that these bodies eventually
undergo a modified form of free fall collapse when self-gravity overcomes magnetic or turbulent
pressures. Thus the rate at which material will fall onto any disk like structure formed in the
collapse is dictated by free fall collapse. The oversimplified isothermal inside out collapse mode of
Shu (1977) provides an order of magnitude estimate for this infall rate:
Ṁin ≈
c3s
≈ 1.6 × 10−6
G
T
10K
3/2
M /yr.
(6)
Somewhat more realistic Bonnor Ebert sphere infall rates are the same order of magnitude, and are
consistent with the timescale on which star formation occurs. In contrast, the observed accretion
rates from disks on to stars in the Class I - Class II phase range from 10−9 − 10−7 M /yr, with the
majority of measurements at the lower end (Gullbring et al. 1998). There are two possible means
to resolve the mismatch between infall and stellar accretion rates. Either disks at earlier times
are more massive, and thus can process more material at higher rate. This option is somewhat
suggestive of self-gravitational effects, but not required. Alternatively if initially disk-star accretion
rates are not higher, the mass of the disk will rise, bringing it ever closer to the regime where
self-gravity is important. It seems unlikely that the infall and disk-star accretion rates, which rely
on entirely different physics (free fall and cloud turbulence vs either self-gravity or magnetic fields),
could conspire to match up at all times in order to maintain a constant, tiny fraction of the total
mass in the disk. Most of the stellar mass is thought to pass through the disk at some point, making
such a balance improbable.
Armitage et al. (2001) first suggested that stellar accretion and FU Orionis type outbursting
systems might be generated by an interplay between gravitational instability driven accretion and
magnetically dominated accretion driven by MRI turbulence. Vorobyov & Basu (2006) suggest that
www.annualreviews.org • GI DISKS
7
GI: Gravitational
Instability
MRI:
Magneto-Rotational
Instability
SPH: Smoothed
Particle
Hydrodynamics
these same outbursts could be triggered by clumps formed due to gravitational instability in the
outer disk that are tidally disrupted as they migrate inwards, and subsequently accrete onto the
star. See the chapter by Hartmann in this Annual Reviews (and recently, the Protostars and Planet
VI chapter, Audard et al. 2014) for more details on variable accretion onto pre-main-sequence stars.
One final piece of evidence for higher disk masses relies on observations of exoplanets. Najita
& Kenyon (2014) have shown that typical protoplanetary disks do not have enough observed dust
mass to account for the mass of planets implied by occurence rates derived from Kepler. They
infer that planet formation, and thus grain growth likely commenced at even earlier times. This
implies that the observed dust masses are missing a substantial amount of solids. Accounting for
this missing mass might well push a larger fraction of disks into the self-gravitating range.
In conclusion we see that while inferred disk masses are often lower than the nominal requirement
for strong self-gravity, it is by no means ruled out. Given the many ways to underestimate disk
masses, the observational evidence demands the study of disk self-gravity .
3. Physics of self-gravity and gravitational instability
In the previous section we have shown that disks may be subject to the effects of self-gravity
when the ratio Md /M∗ > 10−2 . We first address how self-gravity affects the basic hydrostatic
structure of the disk, and subsequently explore how gravitational instability, hereafter GI, may
affect disk structure and evolution. In this section we focus on the physical nature of self-gravity
and GI, which applies to disks in a wide range of astrophysical contexts. We begin by discussing
isolated disks with a fixed surface density profile for simplicity, and subsequently consider the
changes that arise in the presence of infall and external heating. In Section 4, we discuss the extent
to which such physical processes apply to actual protostellar disks.
3.1. Effects of self-gravity on the structure of accretion disks
Consider a thin axisymmetric disk surrounding a protostar of mass M? . Let ρ(r, z), Σ(r) and cs (r)
be the volume density, the surface density and sound speed of the gas as a function of cylindrical
radius r, and height above the disk midplane z, respectively. Standard, non self-gravitating disk
models are based on the assumptions of centrifugal balance in the radial direction and hydrostatic
balance in the vertical direction. Both equations must be modified to account for self-gravity.
A standard non self-gravitating disk in hydrostatic balance is characterised by a Gaussian profile
for ρ, where the disk scale height is given by
Hnsg =
cs
,
Ωk
(7)
p
where Ωk =
GM∗ /r3 is the Keplerian angular velocity and we have assumed the disk to be
vertically isothermal. Self-gravity modifies vertical hydrostatic balance. In the limit where selfgravity dominates, the vertical density profile is given by (Spitzer 1942)
ρ(z) =
ρ0
,
cosh2 (z/Hsg )
(8)
where
c2s
.
(9)
πGΣ
In order to neglect self-gravity in the vertical direction, we require that Hnsg /Hsg 1. This
immediately translates into the condition
Hsg =
cs Ωk
1.
πGΣ
8
Kratter & Lodato
(10)
From this requirement alone we thus recover the usual stability parameter for GI (Toomre 1964).
This means that whenever self-gravity is important in the vertical direction, the disk is near the
instability threshold for GI. This constraint is also equivalent to the constraint on disk mass presented in Equation (2) in Section 2. In the general case where both contributions (of the central star
and of the disk) are important, there is no analytical solution to the hydrostatic balance equation,
but a simple and accurate interpolation formula between the two results above has been found by
Bertin & Lodato (1999) (see also Lodato 2007; Muñoz et al. 2015):
H=
c2s
πGΣ
π
4Q2
#
"r
8Q2
c2
1+
− 1 = s f (Q).
π
πGΣ
(11)
We now consider centrifugal balance. To lowest
p order, centrifugal balance prescribes the angular
velocity of the disk Ω to be Keplerian, Ω(r) = GM? /r3 . Even in the absence of self-gravity, a
small correction is required to account for pressure gradients, which cause the gas to orbit at a
slightly sub-Keplerian speed. This sub-Keplerian motion has a dramatic effect on the dynamics of
small solids in the disk, whose orbits are Keplerian due to the absence of pressure forces. Taking
these corrections into account, we have
s
2
d ln P
H
,
(12)
Ω(r) = Ωk (r) 1 +
r
d ln r
which gives sub-Keplerian rotation because in general P is a decreasing function of radius. Deviations from Keplerian motion due to the pressure gradient are of order (H/r)2 . Self-gravity also
modifies centrifugal balance by a force term which is of the order of GMdisk /r2 (a more detailed calculation, obtained by solving Poisson’s equation, can be found in Bertin & Lodato 1999). Clearly,
in order to dominate the rotation curve, the disk mass needs to be implausibly large for protostellar
disks. However, it is worth noting that when the disk is marginally gravitationally stable (Q ≈ 1),
the disk mass is of the order of (H/r)M? (Equation (2)) and thus the deviation from Keplerian
motion due to self-gravity is of the order of H/r, stronger than the effect of pressure gradients,
though both solids and gas feel the change in Ω(r) due to disk self-gravity.
3.2. Dispersion relations
The most important effect of disk self-gravity on the dynamics of protostellar disks is connected
with the development of GI. The linear stability of self-gravitating disks has been investigated
for decades in the context of galactic dynamics. In the tightly wound approximation, the WKB
dispersion relation for an infinitesimally thin disk is the famous Lin & Shu (1964) relation:
(ω − mΩ(r))2 = c2s k2 − 2πGΣ|k| + κ2 ,
(13)
where ω is the wave frequency, k is the radial wavenumber, m is the azimuthal wavenumber (that
is, the number of spiral arms produced by the instability), and κ is the epicyclic frequency, which,
for a Keplerian disk is simply Ω. We now see that the aforementioned Q from Equation (1) arises
as the threshold for exponential growth of axisymmetric modes, where stability requires
Q=
cs κ
> 1.
πGΣ
(14)
The above dispersion relation and stability criterion have been derived under several important
limiting approximations. The first is that the equilibrium structure of the disk is axisymmetric, the
second (and most important) is that the disk is infinitesimally thin. It can be shown that finite disk
www.annualreviews.org • GI DISKS
9
thickness (Vandervoort 1970; Bertin 2000; Binney & Tremaine 1987) has the effect of stabilizing
the disk by diluting the self-gravity term in Eq. (13). A simple way to account for this effect is to
modifiy eq. (13) as follows:
(ω − mΩ(r))2 = c2s k2 − 2πGΣ|k|e−|kH| + κ2 ,
(15)
that can be expanded to first order to give
(ω − mΩ(r))2 = (c2s + 2πGΣH)k2 − 2πGΣ|k| + κ2 ,
(16)
where now the stabilizing effect of disc thickness becomes apparent, as an additional ‘pressure-like’
term in the dispersion relation. By inserting Eq. (11) in Eq. (16) we finally get
(ω − mΩ(r))2 = (1 + 2f (Q))c2s k2 − 2πGΣ|k| + κ2 .
(17)
From Equation (17) we see that the stability of the disk depends only on the single parameter Q,
and one can show that the marginal stability value decreases to Q ≈ 0.6, for a Keplerian disk.
In the context of galactic dynamics, it was recognized many years ago (Ostriker & Peebles
1973; Hohl 1971, 1973) that disks that are locally stable according to the Q criterion, might still
generate large scale, spiral waves. This is due to the fact that such global modes are not captured
by the WKB tightly wound approximation. However, a WKB description of such modes can still
be obtained under less restrictive conditions than the tightly wound approximation (Lau & Bertin
1978). The resulting dispersion relation is more complicated (it is a cubic rather than quadratic
expression in k) and depends on a new dimensionless parameter J:
J =m
πGΣ 4Ω
rκ2 κ
d ln Ω 1/2 √
Md
≈ 6m
,
d ln r M?
(18)
where the last approximation holds for a Keplerian disk. While the Q parameter balances shear,
thermal pressure and disk mass, J is a measure of the disk-star mass ratio. The cubic dispersion
relation reduces to the standard Lin and Shu expression for the case of light disks (J or Md /M?
much smaller than unity), while for massive disks, where J ≈ 1, or Md /M? ≈ 1/m) lower m, less
tightly wound, spiral arms arise. In both regimes, the most unstable mode scales with H. When H
is small, the behavior is well captured by a local, even 2D analysis. Otherwise, any manifestation
of the instability arises over a significant fraction of the radial extent of the disk.
3.3. The onset of linear instability
When the disk becomes linearly unstable, the onset of GI is similar in both regimes of J. A
spectral analysis of GI in numerical simulations confirms that the dominant modes excited have a
wavenumber k ≈ 1/H, as predicted from the Lin & Shu dispersion relation in Equation (13) (Boley
et al. 2007; Cossins et al. 2009; Michael et al. 2012). Similarly, the azimuthal wavenumber m scales
inversely with H so that thick / massive disks are characterized by fewer spiral arms and by a more
open spiral structure. We provide a more detailed description of the onset of linear instability in
each regime below. Figure 1 illustrates the morphological differences between the instability with
small and large H/r.
Tightly Wound, small H/r. The development of small scale instabilities can be captured by
both local shearing box / shearing sheet simulations and by thin disk global simulations. This
regime is commonly referred to as gravito-turbulence, although some caution should be used when
talking about ‘turbulence’ in this context. Gammie (2001) used a Fourier analysis of GI in the
10
Kratter & Lodato
Figure 1
At left we show a three-dimensional isothermal simulation from the parameter studies of Kratter et al.
(2010a), where Md /M∗ ≈ 0.5. The strong left-right asymmetry is evidence of a dominant m = 1 mode. At
right we show a three-dimensional simulation from Cossins et al. (2009) with slow cooling, where
Md /M∗ ≈ 0.1. Note the dominance of high m spiral structure.
shearing sheet approximation to show that the power spectrum of perturbations generated by the
instability is contained in modes with wavelengths of a few H. Gammie (2001) finds essentially no
power at scales much smaller than H, which are resolved in the calculations. A similar result has
recently been obtained by Shi & Chiang (2014) using three dimensional shearing box simulations.
Young & Clarke (2015) also show – in the context of fragmentation (see below) – that collapse
of unstable modes is never initiated on scales H. These results imply that ‘gravito-turbulence’
extends over a very small range of lengthscales. Additionally, there is no obvious indication of
an energy cascade with dissipation occurring on the smallest scales. On the contrary, in global,
thin disk simulations, Cossins et al. (2009) find that mode dissipation occurs through large scale,
almost sonic shocks. In this respect, the dynamics introduced by local GI is not akin to what one
usually calls turbulence. Morphologically, structures in shearing box simulations look more like
turbulence than the high m, tightly wound spirals seen in global simulations. This might be an
artifact of the shearing sheet/box approximation, but merits further investigation. Whether gravitoturbulence shares properties with isotropic, isothermal turbulence has important implications for
fragmentation, as we discuss in Section 3.8.
Large H/r. As the mass of the disk is increased, the disk becomes unstable at larger values of
H/r and higher values of Q, as predicted from the Lau & Bertin (1978) dispersion relation. Once
the disk-star mass ratio increases above ≈ 0.1, the evolution of the instability changes character
somewhat, transitioning from quasi-stationary spiral structures, to recurrent strong episodes of
spiral activity, followed by brief quiescent phases. This behavior has been observed in a range of
simulations including isolated disks with adiabatic equations of state with and without optically
thin cooling (Lodato & Rice 2005; Mejia et al. 2005; Laughlin et al. 1997), isothermal and irradiated
embedded disks (Krumholz et al. 2007; Kratter et al. 2010a), and even two-dimensional, embedded
disks with radiative heating and cooling (Zhu et al. 2012). A similar, recurrent behaviour has also
been identified in N -body galaxy simulations, where dynamical cooling occurs via the injection of
low-velocity dispersion, or ‘cold’ particles (Sellwood & Carlberg 1984).
www.annualreviews.org • GI DISKS
11
The episodic behavior observed in global simulations is likely due to one of two phenomena.
The first is the growth of the m = 1 mode, as the disk mass becomes comparable to the stellar mass
(Md ≥ 0.3M∗ , in analytic estimates Adams et al. (1989)). Non-axisymmetric perturbations grow
due to interaction with the“indirect potential” (Adams et al. 1989). The indirect potential, caused
by the displacement of the central body from the system center of mass, manifests as an extra
term in the gravitational potential when working in a reference frame centered on the central mass.
These modes arise at higher values of Q than the axisymmetric modes. Simulations of massive disks
that neglect this term may underestimate the growth of instabilities. Under some circumstances,
SLING amplification may occur (Shu et al. 1990), where the m = 1 mode can amplify by interacting
with the disk edge and the outer and inner Lindblad resonances. Because the m = 1 mode leads
to very large mass transport rates, the disk mass drains on such short timescales so that it can (at
least temporarily) become stable and quiescent. The second source of the quasi-periodic behavior
is likely mode coupling, which we discuss in Section 3.5.3.
3.4. Accretion and transport
We now turn to the most important consequence of GI in disks: its ability to drive angular momentum transport, and therefore accretion of matter onto the central object. The fact that a spiral
structure in the disk can transport angular momentum was recognized initially by Lynden-Bell &
Kalnajs (1972) in the context of galactic dynamics. The Rφ component of the associated stress is
given by
Z D
gr gφ E
grav
dz,
(19)
Trφ
=
4πG
where gr and gφ are the radial and azimuthal component of the perturbed self-gravity field and
the brackets indicate azimuthal averaging. To this stress one should also add the induced Reynolds
stress:
Reyn
Trφ
= Σhδvr δvφ i,
(20)
where δvr and δvφ are the perturbed fluid velocities (see e.g. Balbus & Papaloizou 1999). For a
magnetized disk (not considered here), one should also add the relevant Maxwell stress to the two
terms described above. The Shakura & Sunyaev (1973) α prescription relates the stress Trφ to the
local disk pressure by
d ln Ω .
Trφ = αΣc2s (21)
d ln R The dimensionless coefficient, α, often called the anomalous viscosity parameter, is expected to be
less than unity, To facilitate comparison with an eddy viscosity, this is often expressed as
ν = αcs H,
(22)
implying that sub-sonic eddies on scales smaller than H facilitate angular momentum exchange.
The introduction of this parameterization is especially useful for broad parameter studies in a range
disks at moderate computational cost by solving a 1D diffusion equation, often coupled with an
energy equation, as we discuss in Section 3.7. One can also compute numerically the stress induced
by GI (or another process like the MRI) by running a (magneto-)hydrodynamical simulation of an
unstable disk, and compare the resulting total stress to Equation (21) to obtain an effective α.
Let us now consider the torque and power associated with a spiral wave induced by GI. Standard
wave mechanics (Toomre 1969; Shu 1970; Fan & Lou 1999) link the energy and angular momentum
of a wave, Ew and Lw respectively, to the wave action A:
Ew = ωA = mΩp A,
12
Kratter & Lodato
(23)
Lw = mA,
(24)
where Ωp = ω/m is the pattern speed of the perturbation. Thus the ratio of energy and angular
momentum (and the ratio of power and torque exerted on a disk when launching the wave) is the
pattern speed Ωp , rather than the local Ω as for a viscous stress. More explicitly, the wave energy
is given by
2
δΣ
Σ m2
,
(25)
Ew =
Ω
(Ω
−
Ω)
p
p
2 k2
Σ
where δΣ/Σ is the fractional amplitude of the perturbation. Equation (25) can be rewritten as the
sum of two components:
2
δΣ
Σ m2
2
Ew =
(Ω
−
Ω)
p
2 k2
Σ 2
2
(26)
δΣ
Σm
.
Ω(Ωp − Ω)
+
2 k2
Σ
The second term in the above expression is equal to the wave angular momentum times the local
angular velocity and thus represents a local, viscous-like term. The first term is an ‘anomalous’
energy transport term, which is inherently non-local (Balbus & Papaloizou 1999). The ratio of
these two terms, |Ωp − Ω|/Ω, is therefore a dimensionless parameter that measures the degree of
non-locality of the disturbance. In practice, the transport induced by spiral waves will be global
in character if the waves are able to travel within the disk significantly away from their corotation
radius.
We have seen already that the local dispersion relation involving the parameter J (see Equation (18) above) predicts that the transition between local and global behavior is related to the mass
ratio Md /M? . Is the mass ratio condition equivalent to a division of regimes based on the distance
that waves propagate before dissipating? Clearly, to answer this question we have to look at the
spectrum of modes excited by the instability. Using a spectral analysis of the dominant modes
excited by GI, one finds a clear correspondence between morphology and the dominant mode of
angular momentum transport. Cossins et al. (2009) computed the pattern speed of the dominant
modes, finding that independent of the disk-star mass ratio, the Doppler shifted Mach number M
of the dominant perturbation is remarkably close to unity:
M=
|Ωp − Ω|r
≈ 1.
cs
(27)
This implies that the shocks that develop as soon as the waves travel far enough from corotation
to become sonic quickly dissipate the waves. Based on this sonic condition it is now possible to
translate the ‘locality’ parameter defined after Equation (26), into the disk-star mass ratio:
|Ωp − Ω|
cs
H
≈
= .
(28)
Ω
RΩ
r
For a Q ≈ 1 disk, H/r ≈ Md /M? , thus the ‘corotation condition’ and the ‘disk mass’ condition
are equivalent, so long as waves dissipate once they become sonic. We immediately see that the
fraction of transport induced by non-local processes is proportional to the mass ratio.
3.5. Saturation of the Instability vs Non-linear growth
Thus far we have only discussed the linear growth of GI. However, to properly assess the outcome
of the instability we need to consider how it evolves in the non-linear regime. We have introduced
two dimensionless parameters, Q and Md /M∗ that capture the linear phase. In order to understand
the non-linear evolution, we require a third, independent dimensionless parameter, which captures
the radiative timescale of the disk.
www.annualreviews.org • GI DISKS
13
3.5.1. Thermal Saturation. The non-linear evolution of the instability and its saturation is best
understood using numerical hydrodynamical simulations of self-gravitating disks, which have been
the focus of most of the attention in this field over the last twenty years. It is still useful to begin
with a few simple analytic arguments.
Consider first an isolated disk with an initial surface density and temperature profile such that
Q ∼ 1 over a range of radii. In this scenario, Q will evolve either due to mass redistribution or
heating and cooling. We begin by examining changes in temperature rather than surface density
because the viscous timescale is often slow compared to the thermal timescale. Since the excitation of waves will inevitably lead to shocks, it is natural to imagine a self-regulation mechanism
(originally proposed by Paczyński 1978) whereby the parameter Q plays the role of an effective
’thermostat’ for the disk temperature. In the absence of other significant heating terms, the disk
cools down until it reaches marginal stability. At this point, GI sets in, causing shocks that heat
the disk. If sufficient heat is retained, the disk will heat back up, quenching the instability, which
in turn decreases the heating rate, allowing the disk to cool back towards instability. Thus in this
self-regulated state, the disk should remain marginally unstable, with Q close to unity. In such
a state the disk permits growing modes, but the spiral perturbations do not grow exponentially,
instead saturating at some finite value (see Equation (31) below).
One can estimate analytically the conditions under which the disk should self-regulate. Since
the instability provides heat on a timescale that is of order the dynamical timescale in the disk,
tdyn = Ω−1 , the ‘thermostat’ can only work if cooling is slower than this timescale, that is if
β = Ωtcool 1, where tcool is the cooling timescale1 . Thus we see that β is a third, independent
dimensionless number that controls not the onset of the instability like Q, or its morphology, like
Md /M∗ , but its evolution and saturation.
The importance of the parameter β, and the relation to self-regulation, was first illustrated
numerically in a paper by Gammie (2001). This work introduced an explicit cooling term in a
self-gravitating shearing sheet calculation
u
du =−
,
dt cool
tcool
(29)
where u is the internal energy of the fluid and tcool is the cooling timescale, which is set as a
free parameter in the simulations to be some constant times the dynamical timescale Ω−1 . It is
important to stress that the above description is not meant to reproduce any specific cooling law,
but is a toy model for exploring the role of the cooling timescale in the outcome of GI. Different
investigators use different prescriptions for the specific form of the cooling time, which in some cases
is taken to be a constant (Mejia et al. 2005) while in other cases is taken to be proportional to the
dynamical timescale as a function of radius, so that β = Ωtcool is kept constant (e.g., Gammie 2001;
Lodato & Rice 2004; Lodato & Rice 2005). In particular, the so-called β-cooling prescription is
now commonplace when simulating self-gravitating disks in a simplified manner. One noteworthy
limitation of this model is it can only capture the behavior of optically thin gas in numerical
simulations, because it does not account for optical depth effects, which can be important. If
one aims to capture more realistic thermal physics, radiative transfer must be properly accounted
for (Boley et al. 2006; Stamatellos & Whitworth 2009b; Krumholz et al. 2009). Nevertheless, the
evolution produced by simplified cooling models is remarkably similar to that in more complex
simulations.
1 Note that the requirement that the cooling timescale be shorter than the dynamical timescale in order
to result in fragmentation has been discovered in a variety of contexts even outside the context of disk
instability Rees (1976); Silk (1977); Thompson & Stevenson (1988)
14
Kratter & Lodato
In general, there is now a coherent picture emerging from all such simulations of isolated, nonirradiated discs, despite the large variety of hydrodynamic techniques (SPH or grid based codes,
in 2D or in 3D) and cooling prescriptions employed. If cooling is relatively inefficient, so that
the cooling timescale is long (β & 10 − 20, though see Section 3.8), the disk settles into a selfregulated state, where the spiral structure steadily transports angular momentum providing the
heating required to balance cooling and keep the disk in thermal equilibrium. We call this thermal
saturation of GI. We discuss in Section 3.8 uncertainties in the critical value of β and 2D versus
3D effects.
As discussed above, if the disk is low mass (with H/R ∼ Mdisk /M . 0.1), the transport induced
by GI is local, and analogous to a viscosity. If dissipation is the only heating source, then a disk
undergoing GI in thermal equilibrium (Pringle 1981; Gammie 2001), will have the following relation
between α and the cooling rate:
d ln Ω −2
1
.
α = d ln R γ(γ − 1)Ωtcool
(30)
Since the energy density and the angular momentum density associated with density waves scales
with the square of the perturbed gas density (e.g. Eq. 25), the condition for thermal regulation
(Eq. 30) requires that the instability saturates at an amplitude that scales with the square root of
the cooling time:
δΣ
1
∝ √ ,
(31)
Σ
β
a behavior confirmed by the numerical simulations of Cossins et al. (2009). This relation is consistent
with the critical β being of order a few, since it is the point at which density perturbations have nonlinear amplitude. In practice, energy dissipation occurs through the development of roughly sonic
shocks. This simple relationship suggests that the strength of the instability, and its efficiency as
an angular momentum transport process, are correlated with the thermodynamics in steady state.
We review GI models that use α as a proxy for transport in Section 3.7.
3.5.2. Thermal saturation in irradiated disks. The discussion and the simulations described above
assume that the only source of heating in the disk is that provided by the instability itself as it
dissipates energy through shocks. However, protostellar disks are generally expected to be irradiated by either the central protostar, their natal envelope, or nearby stars. In the presence of
irradiation, clearly the thermal saturation described above must change, since the disk alone is not
responsible for setting the equilibrium temperature. The thermostat cannot function according to
Equation (30).
When comparing GI in irradiated versus self-heated disks, one must consider the behavior at
fixed Q, so that the disk has the same propensity to the instability. Even at fixed Q, the impact
of irradiation is not intuitive. On one hand, in the presence of external heating a much lower level
of dissipation due to the instability is required to prevent runaway growth, which would imply a
stabilization of the disk. On the other hand, a very strongly irradiated disk will behave more like
an isothermal disk, where the dissipation of energy due to even a large perturbation has little effect
on thermal balance (Kratter & Murray-Clay 2011). In this case it is hard to envision a mechanism
that would stop an unstable perturbation from growing non-linearly, leading to fragmentation.
Indeed irradiated disk simulations have more similarities to isothermal simulations than barotropic
or cooling simulations (Krumholz et al. 2009; Offner et al. 2009). For disks in the moderately
irradiated regime, both arguments apply. Rice et al. (2011) conducted local 2D simulations of
disks including a ‘standard’ β-cooling term and a constant heating term, representing the effects of
irradiation. On the basis of cooling times, irradiation stabilizes the disk in that a marginally faster
www.annualreviews.org • GI DISKS
15
cooling time is required for fragmentation in the irradiated case as compared to the non-irradiated
case. However, Eq. 31 does not hold anymore, and for a given value of β the average perturbation
amplitude is much smaller. Thus, in the irradiated case, even perturbations with a very small
average amplitude result in a runaway collapse into fragmentation. In this sense, irradiation has
the effect of destabilizing the disk, because smaller amplitude perturbations can collapse. It is
unclear whether the use of 2D simulations in this study might affect the measured critical β (Young
& Clarke 2015).
In irradiated disks, there is another route to saturation that does not relate to thermal balance,
but rather is dynamical, in the form of non-linear coupling between modes.
3.5.3. Mode-mode coupling. The numerical investigation of self-gravitating disks began well before
sophisticated radiative transfer cooling algorithms were computationally feasible. Early numerical
simulations found that both isothermal (Laughlin & Bodenheimer 1994), and barotropic (Laughlin
& Ròżyczka 1996; Laughlin et al. 1998) disks could develop strong gravitational instabilities that
lead to fast redistribution of angular momentum within the disk. In these regimes, GI cannot
self-regulate in the manner describe above as they represent β → 0 and β → ∞ respectively.
Instead, non-linear mode coupling appears responsible for saturating the amplitude of GI-induced
perturbations. Laughlin et al. (1997) demonstrate that coupling between m = 0 and m = 2
modes prevents runaway growth and thus fragmentation, while allowing rapid angular momentum
transport. This same behavior is evident in the isothermal simulations of Kratter et al. (2010a),
and is also likely responsible for the transient spiral patterns seen in the massive disk simulations
of Lodato & Rice (2005).
The exact conditions under which mode-mode coupling dominates remain unclear, as little
detailed work has been done since Laughlin et al. (1997). Based on the simulations in which a
similar effect has been observed, we expect this saturation mechanism to dominate in massive
disks, as well as disks that are irradiation dominated. These two cases represent (1) the regime
in which local dissipation breaks down and (2) the regime in which the energy balance due to GI
induced heating and radiative cooling breaks down. Ironically, this saturation mode may be the
dominant mechanism preventing fragmentation in protostellar disks, despite the dearth of attention
in the literature.
3.6. When saturation fails: Fragmentation
If the saturation mechanisms listed above fail to prevent runaway growth of perturbations, the
unstable region of the disk will break up into bound objects, or fragments. As the perturbations
grow to non-linear amplitudes, their own self-gravity causes them to collapse and separate from
the background disk. Fragmentation is in some ways an alternative means for shutting off the
instability: the separation of large amplitude perturbations form the disk means a reduction in
local surface density, and thus an increase in Q. In the context of protostellar disks this is of great
importance, as fragmentation can in principle lead to the formation of either stellar, brown dwarf,
of perhaps planetary mass companions. We provide a brief overview of the two common pathways
to fragmentation, deferring a discussion of the astrophysical implication to Sections 4 and 5
Fragmentation driven by cooling. A great majority of disk simulations have considered disk
fragmentation as consequence of rapid cooling in isolated disks. In this case, Q decreases due to
a progressive drop in temperature, until GI sets in. If the cooling is rapid, GI cannot provide
sufficient heating to counteract cooling, and the disk will fragment. Exactly how fast the disk must
cool (the so-called critical value of β) has been the subject of many investigations. Studies have
16
Kratter & Lodato
examined the role of the temperature dependence of cooling (Johnson & Gammie 2003; Cossins
et al. 2010a), the role of the evolution of β in time at fixed radius (Clarke et al. 2007), and the
role of irradiation (Boley et al. 2007; Stamatellos & Whitworth 2008; Rice et al. 2011). The critical
value is a function of the equation of state, and is of order β = 10 − 20. The deviations from the
critical value of β due to more complex physical effects are of order a factor of two. We review the
numerical complications of measuring this value in Section 3.8.
Cooling as a path to fragmentation is most likely relevant for disks that are thermally selfregulated because, as discussed above, irradiation and global transport can hamper the operation
of the disk thermostat. AGN disks, or protostellar / protoplanetary disk which are shadowed by
inner disk wall or warps may be the best candidates for this type of fragmentation. Alternatively
disks which undergo rapid changes in accretion rate, which in turn can effect the stellar luminosity
and thus the irradiated disk temperature, might also be subject to fragmentation driven by cooling
(Kratter & Murray-Clay 2011).
Fragmentation driven by accretion. The second path to fragmentation is due to evolution of
the disk surface density either due to infall from the external environment, or due to changes in
the accretion rate within the disk that lead to mass pile-ups. For disks undergoing infall at some
rate Ṁin , there exists a regulation mechanism akin to thermal saturation. Consider a disk with
Q > 1 that accretes material such that Q declines towards unity. As the disk approaches Q ≈ 1,
the linear instability transports angular momentum. If the accretion induced by GI gives Ṁ < Ṁin ,
the disk will become more massive, and thus susceptible to a wider variety of modes, enabling more
transport. If Ṁin ≤ Ṁmax,GI , the maximum possible rate that GI can provide (as yet unspecified,
see below), the disk will transport material at the rate it is being fed, with the disk mass acting
as the regulator of transport (Kratter et al. 2010a; Zhu et al. 2012). If Ṁin > Ṁmax,GI the disk
will ultimately fragment (Kratter et al. 2008). The ability of the disk to regulate transport up to
a critical rate is likely related to the spiral-mode coupling described above.
What sets the maximum transport rate induced by GI, and how does one know when disks will
fragment versus increase in mass? Kratter et al. (2010a) argued that two dimensionless parameters,
benchmarked to the infall rate, are strong predictors of fragmentation driven by accretion. These
two are
ξ = Ṁin /(c3s /G)
(32)
Γ = Ṁin /(M∗d Ω),
(33)
where M∗d , the total system mass. The first parameter, ξ references the infall to the isothermal
sphere collapse rate, c3s /G (Shu 1977) but uses the disk, not core temperature. For a disk in steady
state, ξ and α are interchangeable, using the steady-state accretion relation Ṁ = 3αc3s /(GQ)
(Frank et al. 2002). However in many cases, the disk accretion rate does not come into equilibrium
with the infall rate. The second parameter, Γ is the ratio of the orbital timescale to the disk
mass doubling timescale. When infall is significant, Ω is controlled by the circularization radius of
accreting material (not viscous spreading) and is thus a proxy for disk angular momentum.
Using a suite of 3D numerical simulations of isothermal disks undergoing self-similar accretion,
Kratter et al. (2010a) find that disks undergoing infall with Γ < ξ 2.5 /850 are unstable to fragmentation. Subsequent work by Offner et al. (2010) and Zhu et al. (2012) confirm these relations hold
when more complicated thermal physics and more realistic, turbulent infall conditions are included.
Such a parameterization is applicable to AGN disks as well, where it might be relevant for seed
black hole growth at high redshift (Lodato & Natarajan 2006).
The ξ, Γ parameterization predicts that the maximum transport rate is a function of mass
ratio (in steady state with Q ∼ 1, we have that H/R ≈ Md /M∗ ≈ (Γ/ξ)1/3 ). For low mass disks,
www.annualreviews.org • GI DISKS
17
where GI-induced transport is local, Rice et al. (2005) have shown that the existence of a critical
cooling rate for fragmentation translates, through Equation (30), into a more general upper limit
of α ≈ 0.1. The boundary for fragmentation in ξ − Γ space is consistent with this value of α
for equivalently low mass disks, despite the absence of thermal regulation. If the disk mass grows
above 0.1−0.2M∗ , however, transport becomes inherently global, and thus an increase in mass ratio
corresponds to an increase in the maximum rate of transport. This is consistent with the Lau &
Bertin (1978) dispersion relation. According to the ξ − Γ boundary derived above, a more massive
disk corresponds to a larger value of Γ for fixed ξ, which will thus be stable at the same infall rate.
These simulations suggest that global modes can provide close to α ≈ 1 when Md → M∗ .
Other fragmentation triggers. Nelson (2000); Mayer et al. (2005); Boss (2006) and Lodato et al.
(2007) have studied the influence of a massive companion on disk fragmentation, reaching apparently
contradictory conclusions. While Boss (2006) argues that the presence of a companion increases
the tendency for fragmentation, all of the other studies obtain the opposite result. They find that
the tidal heating due to the companion has the effect of inhibiting fragmentation in a marginally
stable disk.
3.7. Capturing GI with α models
As mentioned in the previous sections, it is often convenient to parametrize angular momentum transport measured in numerical simulations in terms of the Shakura & Sunyaev (1973)
α-parameter. We have relied on this parameterization heavily in discussing thermal saturation
(see Equation (30)). Despite the inherent assumption of local transport in the α-model (Balbus &
Papaloizou 1999), the approximation is also often used for global transport, where it approximates
the accretion rates measured remarkably well. Kratter et al. (2008) compiled transport measurements from numerous simulations, and found that a two component α model – one to account for
local transport, one for global, could accommodate a wide range of disks in terms of both mass and
thermal physics. As noted above, the biggest advantage of invoking an effective α to capture GI is
that one can conduct parameter studies over a wide swath of physical variables using semi-analytic
or 1D models. We provide a brief overview of the most common models used.
The first proposed αsg models of Lin & Pringle (1987, 1990) posit that since Q controls the
onset of instability, the strength of transport should scale inversely with Q. They propose:
αsg
 2
Q̄


−
1
ζ
Q2
=



0
Q < Q̄
(34)
Q > Q̄
where ζ is an additional parameter measuring the strength of the gravitationally induced torque, and
Q̄ is the maximum value (≈ 2) at which GI sets in. This prescription, at face value, is orthogonal to
α models based on Equation (30). Although the dependencies appear distinct between Equation (30)
and Equation (34), they can be tuned to capture similar behavior, but are better suited to different
regimes. Equation (30) is appropriate for disks that are in steady-state and thermally self-regulated.
It cannot be used to capture the behavior of a disk with Q much different from unity, or one that
is out of equilibrium because the relation is predicated on energy balance. On the other hand,
Equation (34) can capture GI in a disk that is non-self regulated and may be evolving (either in
surface density or temperature) in time. Because Equation (34) has a free normalization parameter,
it can also describe a disk in steady-state that is marginally unstable by tuning ζ so that the
transport rate matches up with the R.H.S. of Equation (30).
18
Kratter & Lodato
These α models have proved indispensable for studying multiple transport processes simultaneously. Armitage et al. (2001) explored 1D viscous models including both MRI and GI-induced
transport, showing time variable accretion capable of reproducing FU-Ori like episodic accretion.
Bertin & Lodato (2001) discuss how GI would make the outer disk in FU Ori objects hotter compared to non self-gravitating disks and hence produce a flatter SED (see also Adams et al. 1989).
Martin & Lubow (2011) similarly identify regions of disk parameter space lacking steady-state solutions as responsible for outbursting behavior, in a fashion akin to the limit-cycle instability for
dwarf novae. Matzner & Levin (2005) first identified, through simple α models, the existence of
a critical radius at ≈ 50 − 100 AU beyond which the disk is expected to fragment. Clarke (2009)
use an α model for GI to estimate the relevant cooling time as a function of radius, and show that
fragmentation occurs beyond ∼ 70 AU, while self-regulation is expected between ∼ 20 − 70 AU.
Crucially, Clarke (2009) neglected the effect of irradiation from the central star, that in general
makes the inner disk gravitationally stable rather than self-regulated.
Recently, Rafikov (2015), following Rafikov (2009), developed a more comprehensive viscous α
prescription, which encompasses a wider range of disk states, although the caveats about locality of
energy deposition remain. Rafikov (2015) finds that the predicted disk properties are quite similar
to those derived under α(Q) models (e.g. Kratter et al. (2008); Zhu et al. (2010)). The important
feature of these viscous models is the smooth growth of transport rates as Q declines, with a cutoff
at some maximum α for a minimum Q.
3.8. Convergence of numerical results
In our analysis of the saturation of GI, we have relied heavily on numerical simulations. The two
values often pulled from these simulations are the critical value of β, or the maximum transport rate
α, at which saturation fails, leading to fragmentation. These two quantities (directly correlated in
some regimes) are only measurable in simulations. Frustratingly, these quantities can be difficult
to disentangle from numerical artifacts. No astrophysical simulations operate at realistic Reynolds
numbers, and all suffer from numerical errors that manifest as an effective viscosity. In the case of
SPH simulations, an artificial viscosity must be included explicitly. For grid codes, Krumholz et al.
(2007) for example, has measured the effective α from diffusion as a function of resolution, showing
that it can be competitive with physical mechanisms at moderate resolution.
At the time of this writing, the convergence problem in self-gravitating disks simulations is
an open issue. We review them below for completeness, but emphasize that for application to
protostellar and protoplanetary disks, uncertainty in the critical value of β has little impact on disk
evolution. Indeed, in the region where protostellar disks are gravitationally unstable, the cooling
time is a steep function of disk radius, so that changes in the critical value of β only lead to minor
changes to the radius at which fragmentation occurs (Clarke 2009; Clarke & Lodato 2009).
Until recently, most investigations of resolution focused on resolving fragmentation. Early
studies found that the Jeans length (Bate & Burkert 1997; Truelove et al. 1997) must be resolved
by ≈ 10 grid cells to accurately capture fragmentation. More recent work has argued that the
Jeans length must be resolved by as many as 64 grid cells to achieve convergence (Turk et al. 2012).
Since in SPH the resolution is set by the smoothing length h, and the Jeans length is of the order
of the disk thickness H, the Truelove and Bate & Burkert criteria imply that h/H . 1 to resolve
fragmentation. This condition is amply satisfied by most recent simulations. Nelson (2006) lists
three different conditions: one is a variant of the Jeans length criterion, the second is that h/H . 1,
and the third is that adaptive smoothing lengths be used in SPH (which is done in all modern SPH
calculations). That convergence remains a challenge even when the above resolution criteria are
met is not so surprising, since none explicitly capture one of the most important aspects of the
www.annualreviews.org • GI DISKS
19
physics: dissipation.
Meru & Bate (2011a,b) found that in simulations with several million SPH particles (factors of
several above most previous work) the fragmentation criteria as measured by a critical β did not
appear to converge. The disk fragmented at larger β with higher resolution. Similar results have
come from grid-based simulations. Paardekooper et al. (2011) found that fragmentation could be
artificially triggered at sharp boundaries between the stable and unstable portions of the disk, while
Paardekooper (2012) claimed, based on 2D simulations, that fragmentation might be a stochastic
process, and therefore allowed (rarely) at values of β associated with much smaller amplitude
perturbations. This interpretation would be consistent with gravito-turbulence fully sampling a
turbulent power spectrum of density perturbations (Hopkins & Christiansen 2013). These results
cast doubt on the numerical value of the fragmentation boundary obtained in previous investigations
(that pointed to a critical value βcrit ≈ 10).
Lodato & Clarke (2011) address the importance of the magnitude of the artificial viscosity as a
function of resolution, which inherently alters the disk thermostat. In SPH codes, artificial viscosity
provides an equivalent αart ∝ h/H. Lodato & Clarke (2011) find that in order for artificial viscosity
to be negligible as a source of heating, one requires
40
h
.
H
β
(35)
Note that as β is increased (as one does when probing the fragmentation boundary), the resolution
requirement becomes progressively harder to satisfy, because a smaller amount of numerical dissipation represents a larger fractional contribution to heating. Such a trend of increasing resolution
requirements with increasing β is indeed consistent with the Meru & Bate (2011a,b) results, but
do not fully explain them, as it turns out that, based on these simple arguments, fragmentation
would be artificially quenched by numerical viscosity when the latter produces only 5% of the heat
required by thermal equilibrium. Meru & Bate (2012), on the other hand, point out that most SPH
simulations for self-gravitating disks have employed an artificial viscosity formulation that strongly
reduces viscosity for high Mach number shocks. Paradoxically, this reduction of the viscosity coefficient has the counter-intuitive effect of increasing artificial dissipation, since it generates artificially
large velocity gradients (Lodato & Price 2010).
Even more extreme challenges to the fragmentation boundary have come from Paardekooper
(2012) and Hopkins & Christiansen (2013) who suggest that fragmentation may be inherently
stochastic, and thus allowed at much longer cooling times. This relies on GI producing a wide
power spectrum of turbulent fluctuations Investigations by Young & Clarke (2015) question this for
two reasons. First, they show that independent of numerical method, there are inherent limitations
in the treatment of the small scale gravitational field in 2D. As a result, measurement of the
fragmentation boundary (or stochastic fragmentation) in a 2D simulation is questionable, and
better left to 3D simulations. They further argue that because of the quasi-regular nature of selfgravitating structures, stochastic fragmentation should be inhibited at very large β, because the
growth and development of spiral structure occurs over a modest number of orbital periods, and
thus may not be akin to a true turbulent cascade (see also Shi & Chiang (2014)).
Despite the lingering questions about resolution and critical cooling times, we emphasize that
the critical β for fragmentation has very little impact on whether disk fragmentation, and especially
planet formation via GI, is viable because most disks that have Q ∼ 1 also attain relatively small
values of β for realistic protostellar disk conditions.
20
Kratter & Lodato
4. Application to Protostellar Disks
In Section 3 we have given an overview of the basic physics that governs the behavior of disks when
self-gravity becomes important. These principles apply to disks in any astrophysical context. Here
we narrow our focus to the case of protostellar and protoplanetary disks in order to understand which
of the regimes and corresponding behaviors are most relevant. The details of how star formation
proceeds inevitably effects the characteristics of protostellar disks, and thus the importance of
self-gravity and GI. It is beyond the scope of this review to describe how different modes of star
formation might produce different disk properties; instead we endevaor to provide a broad overview
of all plausible disk states.
Narrowing the Parameter Space. We have shown that GI can be described by three independent
dimensionless numbers that describe the state of the disk, Q, Md /M∗ and β. However, a protostellar
or protoplanetary disk does not attain arbitrary combinations of these three quantities because the
disk’s internal energy and cooling rate are both temperature dependent. Consider β in terms of
these physical variables:
Σc2s
4
τ
(36)
tcool = βΩ−1 =
9γ(γ − 1) σT 4
where τ = κΣ/2, and κ is the opacity. The cooling time, and thus β, depend on the disk surface
density, temperature, and dust opacity. Once one selects a disk model with a given Q profile and
mass ratio, the profile β(r) is also determined (albeit with uncertainty due to dust opacity models).
We present typical disk scalings for temperature, mass, size, and accretion rate in order to survey
GI in protostellar and protoplanetary disks, highlighting three case studies.
4.1. Setting the Thermal Physics
We begin with a simple model for the temperature profile of the disk. Ideally disk temperatures
should be found from accurate radiative transfer calculations, which account for wavelength dependent opacities and heating sources. Nevertheless, the agreement between the simple analytic
formulae below and radiative transfer calculations is quite good, making these useful for estimation
purposes (Zhu et al. 2012; Boley et al. 2010).
Disks are heated primarily by three sources: accretion energy, host star irradiation, and external
radiation. The disk midplane temperature can be found via the following relation by balancing
heating and cooling terms:
3
σT 4 = f (τ )Facc + F∗ + Fext
(37)
8
where F∗ and Fext are the energy fluxes associated with stellar irradiation and external heating
respectively, while Facc is the energy flux associated with accretion and
f (τ ) = τ +
1
τ
(38)
(Rafikov 2005). Here, τ is the Rosseland mean opacity, and f (τ ) reasonably captures how the
accretion energy diffuses outward from the midplane in both the optically thick and thin regimes.
We now examine each term in detail.
The first term on the R.H.S, Facc , describes the release of gravitational energy as material moves
through the disk, falling deeper into the potential well. This is commonly referred to as viscous
dissipation. Independent of whether the disk is truly viscous, the potential energy of accreted
material must be released, some portion of which will go into the thermal energy of the disk. The
flux (far from the stellar surface) associated with this term if all of the accretion energy goes into
www.annualreviews.org • GI DISKS
21
heating is:
3
Ṁ Ω2
(39)
8π
Because the heat must propogate from the midplane to the surface, this term is modified by the
optical depth. As discussed in Section 2, disk opacities are uncertain, and likely evolve in time due
to grain growth and changes in the dust-to-gas ratio. At very early times, when grains are ISM-like
(µm − 10µm), the opacity is temperature dependent, typically κR ≈ κ2 T 2 (where T is in Kelvin),
(Bell & Lin 1994), in the cold, icy limit T < 155K relevant for GI. Semenov et al. (2003) find that
the Rosseland mean opacity for small grains is well fit by κ2 ≈ 5 × 10−4 cm2 /g. Once grain growth
commences, grains quickly become large compared to the blackbody wavelengths of the disk. In
this regime, the opacity is both lower and temperature independent. Based on Pollack et al. (1985)
we use a temperature independent κ0 = 0.24cm2 /g. We benchmark our results to this large grain
case for several reasons. First, after no more than 105 yrs, there is good evidence that grain growth
has commenced. Although GI may be present earlier, at this epoch the disk is optically thick for
either opacity law. Secondly, lower opacities associated with grain growth produce a wider range
of disks susceptible to GI; since our intent is to review all the possible phase space we consider this
choice “conservative.”
The second term on the R.H.S. of Equation (37) captures irradiation from the host star. The
stellar luminosity comprises two components: accretion luminosity and the intrinsic stellar luminosity that comes either from gravitational radiation or nuclear burning depending on age and mass.
For low mass stars the accretion luminosity typically dominates. High mass stars, which reach the
ZAMS while accreting, are dominating by H-burning (Tout et al. 1996). Unlike the dissipation of
accretion energy, the impact of stellar irradiation on disk temperatures is highly dependent on the
envelope and disk structure, because the absorption of radiation depends on both the intervening
material reprocessing the radiation and on the flaring angle of the disk absorbing radiation. We
consider two models for stellar irradiation. The first, from ray tracing calculations of Matzner &
Levin (2005), finds that for deeply embedded disks, the heating term is well described by:
Facc =
4
F∗,emb = σT∗,emb
=f
L∗
4πr2
(40)
with f ≈ 0.1 derived from ray tracing calculations where much of the stellar flux impacts the
infalling envelope (ala Terebey et al. (1984)) and is reprocessed back down onto the disk. Note that
L∗ can include both internal luminosity and stellar accretion luminosity. The second model refers
to the case of a non-embedded disk, for which the analytic model of Chiang & Goldreich (1997)
accounts self-consistently for the flaring due to incident irradiation, and heating of dust grains in
the outer disk layers. They find:
α R 2
F
∗
4
F∗,iso = σT∗,iso
=
(41)
σT∗4
4
r
where αF measures the grazing angle at which starlight hits the disk; this in turn is a function
of the height of the disk photosphere, which depends on grain properties. When disks reach the
threshold for instability, they are typically optically thick, making the use of this approximation
reasonable. We can rewrite this explicitly as a function of stellar luminosity and disk radius as:
1/7
kb 1
1
−3/7
T∗,iso =
L2/7
(42)
∗ r
(7πσ)2 µ GM∗
We note that when using this relation for temperature due to stellar irradiation in conjunction with
other heating terms, there is a small inconsistency in that the actual disk scale height will differ
slightly from that solved for in the irradiation model.
22
Kratter & Lodato
Including both heating terms, we see that an individual disk may span both the self-luminous
and irradiated regime at different radii. It is instructive to examine the radial scaling of the first
and second terms in Equation (37):
Facc ∝ Ṁ (r)r−3
(43)
F∗,emb ∝ L∗ r−2
(44)
−12/7
L8/7
∗ r
(45)
F∗,iso ∝
where the last two lines cover the embedded and isolated cases. In either case, the viscous term will
fall off more steeply with radius than either irradiation term. Thus the inner disk is more likely to
be self-luminous, while outer disk may be dominated by irradiation from either the central star or
even the external environment.
Finally, external radiation from other stars can contribute to the disk temperature (see Thompson 2013 for an extreme example). We consider the interstellar radiation field to create a tempera4
ture bath of Text (such that Fext = σText
), which sets a lower threshold for the disk temperature as
described above. In more active, higher mass star forming regions this can be up to Text = 20−30K
(Caselli & Myers 1995). For deeply embedded cores, the envelope may insulate the disk from external sources, providing a somewhat lower temperature thermal bath. Neither irradiation term is
accompanied by an optical depth because they set the temperature at the surface, which controls
the cooling (L.H.S. of Equation (37)).
4.2. Disk Masses and Radii
In Section 2 we showed evidence for disks ranging from Md /M∗ ≈ 0.001 − 0.1, but argued that
in many cases these may be lower limits based on uncertain grain growth, dust-gas conversions,
and biases against observing the youngest, most massive disks. Here we consider disks with 0.05 <
Md /M∗ < 0.5 to encompass both the least massive disks where self-gravity can be relevant, and
a generous upper limit on those observed. Recalling Equation (2), one can immediately see that
a Q = 1 disk at the upper boundary has an aspect ratio so large as to call into question the disk
geometry.
Direct observations show that disks extend out to 50 − 500au, based on submm and optical
absorption of proplyds (Andrews et al. 2009; Eisner et al. 2008). Disk radii at early times (Class 0,
I phases) are controlled by the infalling angular momentum from the envelope or feeding filament
(Terebey et al. 1984). Most of the disk mass will be contained within the circularization radius:
hji2
rd = R c = √
GM
(46)
where hji is the average specific angular momentum of accreting material, which can be obtained
from considering
J̇(t) = Ṁ (t)j(t).
(47)
In a simple core-collapse model one can specify j(t), however it is more realistic to consider the
impact of accretion from turbulent filamentary structures that characterize star forming regions
(Offner et al. 2010). Turbulence in the interstellar medium follows a line-width size relation, such
that increasingly larger scales will typically carry more angular momentum (Larson 1981). Kratter
& Matzner (2006) used this to generate models for disk radii as a function of time. It is not
guaranteed, however, that the direction of the angular momentum vector will remain constant
throughout infall, and thus disks may not grow monotonically in time. Recent simulations by Bate
et al. (2010) and Fielding et al. (2015) show that there can be enough change in the infalling angular
www.annualreviews.org • GI DISKS
23
momentum vector to tilt the disk with respect to the spin-axis of the star, especially if variable
accretion causes substantial oscillations in the disk mass, and if stellar interactions truncate the
disk.
When disks transition from the protostellar to protoplanetary phases, infall of mass and angular
momentum becomes less important. In this case one expects the disk to evolve towards the selfsimilar viscous spreading solution of Lynden-Bell & Pringle (1974). While this will lead to larger
outer disk radii than that given by the circularization radius, the outer regions typically hold very
little mass and thus do not have a significant impact on disk self-gravity.
In order to determine Σ(r), we require a model for the radial profile. Here again we rely on
observations, which suggest a power law, or a power law plus exponential tail (Andrews et al.
2009; Pérez et al. 2012). Since the exponential part contains an irrelevant amount of the mass, we
consider only the power law component. For our calculations, we choose Σ ∝ r−1 , which puts more
mass at large radii where instabilities might arise, lest we underestimate the role of self-gravity.
Note that this surface density profile, coupled with an irradiation dominated temperature profile
is consistent with steady-state disk models with Ṁ (r) =const. To give a quantitative idea of the
scaling for our parameter ranges:
Σ(r) = Σ0
r
Rout

−1  Σ0 = 28g/cm−2

,
Md
M∗
= 0.05, Rout = 50au
Σ0 = 2.8g/cm−2 ,
Md
M∗
= 0.5, Rout = 500au
(48)
4.3. Accretion Rates
The only remaining parameter to specify is the accretion rate through the disk, which has two effects.
First, as discussed in Section 3.6, infall of material on to the disk impacts the disk mass, angular
momentum transport rate, and susceptibility to fragmentation. Secondly, the disk temperature is
a function of the accretion rate. We consider disks with steady-state accretion at rates ranging
from Ṁ = 10−8 − 10−3 M yr−1 , where the lower end is consistent with accretion rates observed
on actively accreting T Tauri stars towards the end of their lives, and the highest rates represent
a generous upper limit set by infall from a supersonic core for more massive stars (McKee & Tan
2003; Klaassen et al. 2011) .
In reality, a disk with a given surface density and temperature profile cannot support an arbitrary accretion rate in steady state. A physical mechanism for providing angular momentum
transport is required. Indeed there is no guarantee that a disk will typically be in steady-state with
Ṁ (r) = const (Armitage et al. 2001; Martin & Lubow 2011).
As discussed in Section 3, saturated GIs provide transport at rates such that 10−2 < α < 1
depending on the disk parameters. Obviously magnetic effects, both small scale turbulence from
the MRI (Balbus & Hawley 1994; Simon et al. 2013), or large scale magnetic winds (Bai 2015) may
also be important for various disk properties at different times.
4.4. The instantaneous disk state
We demonstrate how GI manifests in protostellar and protoplanetary disks with three case studies
shown in Figure 2 that cover a range of parameters. Note we have not yet specified how disks might
evolve into and out of these parameters, and some areas of the parameters space are unrealistic.
For each case study we provide two contour plots of fundamental disk parameters as a function of
disk radius and accretion rate. We fix the disk mass, outermost radius and host star properties,
and then calculate the implied properties of the disk at all locations for a range of accretion rates.
The left hand panels show contours of the disk temperature with all three heating terms from
24
Kratter & Lodato
Equation (37). For comparison we also show in white contours the temperatures that would be
derived neglecting stellar irradiation, as this has been omitted in many numerical simulations. The
accretion rate not only contributes to the viscous heating term, but also the stellar heating term,
where for low mass stars we use the models of Baraffe et al. (2015) to estimate the stellar radius
at 1Myr. The right hand panels show contours of Q and β (calculated from Equation (36)). When
relevant, we demarcate the boundary where the accretion rate assumed would require α > 1. This
region is inconsistent with GI as a transport mechanism.2 We describe each of the three cases in
detail below:
2 Note that the slope of the constant α contours, indicated by the boundary of α > 1 shaded region, are
close to parallel with lines of constant Ṁ . This suggests that the chosen temperature and density profiles
are reasonably consistent with steady-state, constant α disks.
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25
Figure 2
Three case studies for protostellar disk properties as functions of accretion rate and disk radius. The left
panels show the disk temperature including (black) and excluding (white) stellar irradiaiton. The right
panels show the value of Q (background color and black contours) and β (white contours). The stellar
masses and disk masses used are shown in the upper left corner of the lefthand panels. For Disk 1, note
that the shaded gray contours indicate solutions which require α > 1 and thus are not physical.
Disk 1. : In Figure 2a we show results for Md /M∗ = 0.05, Rout = 100au, around a solar mass star.
This disk is most consistent with observations of Class II protoplanetary disks. It is immediately
clear that these disks will not be subject to GI at any radius. Q is lowest at the outer edge, where
cooling times are also very rapid, and the temperature is controlled by stellar irradation. In order
to make the outer regions of this disk unstable to GI, the disk mass must be increased by roughly a
factor of 4. In this case the cooling time remains short, suggesting that such disks would fragment,
but never enter the self-regulated regime, as first recognized by Rafikov (2005); Matzner & Levin
(2005).
Disk 2:. Figure 2b shows a disk modelled after observations of IRAS 16293-2422b (Rodrı́guez
et al. 2005). This disk is most likely associated with an isolated Class 0 source. The disk appears
to be massive and compact, consistent with a very young age, with Md /M∗ = 0.35, Rout = 30au
around a 0.8M protostar. The Rodrı́guez et al. (2005) model suggests an accretion rate of order
10−6 M /yr, but we show the full range as for the other examples. Because the disk has low Q and
is slowly cooling, we expect one of two outcomes depending on the rate at which mass is accreted
onto the disk. Up to some critical rate, the disk can process the material via GI-induced transport
and remain quasi-stable. This would require a combination of thermal regulation and mode-mode
coupling. At larger infall rates, the disk will begin to grow in mass, and either revert to the first
state of self-regulation, or increase in mass until it fragments.
Disk 3. Figure 2c shows a disk model appropriate for a massive star, such as that identified by
Shepherd et al. (2001), consistent with theoretical models of high mass star formation (Kratter
& Matzner 2006; Krumholz et al. 2007). Here Md /M∗ = 0.75, Rout = 150au and M∗ = 8M .
Unlike the low mass stars in the previous two examples, high mass stars are expected and observed
26
Kratter & Lodato
(Klaassen et al. 2011) to have higher infall rates. Therefore the upper half of the plot is more
representative of likely parameters. Here we see that the entire outer logarithmic decade in radius
will have Q ∼ 1. While at distances of the order of tens of au the cooling time is relatively long
and the disk might survive in a self-regulated state, beyond ∼ 50au the cooling time becomes short
enough to induce fragmentation. The disk temperatures are higher in part because massive stars
arrive on the ZAMS and start fusion while still accreting, but they still have low Q values. The
high mass ratio represented in this example may be typical because the high infall rates tend to
drive up disk masses. In this case, simulations suggest that disk undergo strong m = 1 dominated
transport, and in many cases fragmentation into stellar companions (Adams et al. 1989; Kratter
et al. 2010a). The susceptibility of these disks to fragmentation is consistent with the observation
that nearly all massive stars are found in binaries (Sana & Evans 2010; Sana et al. 2014).
4.5. Global Models for Disk Evolution
We have laid out the instantaneous properties of protostellar and protoplanetary disk in the previous
sections. However, protostellar disks do not exist in isolation, but are constantly fed material from
the background core or filament. We argued in Section 3 that infall can control the linear and
non-linear development of GI. There, we simply imagined that a given disk, and considered what
happened as we changed the infall rate. In order to understand what properties disks have during
their evolution, star formation models are required that dictate Ṁin and J̇in . The state of the disk
is controlled by competition between the infall rate and the disk-star accretion rate.
Several authors have considered this global evolution: Kratter et al. (2008), who used analytic
models tied to numerical benchmarks for non-linear transport terms, Vorobyov & Basu (2007, 2009),
who consider 2D models for the disk considering only transport driven by GI, Zhu et al. (2010,
2012), who consider long timescale semi-analytic models, and shorter timescale 2D simulations.
These calculations suggest that low mass stars, as demonstrated by disk case studies (1) and (2),
are susceptible to GI at early times, but stable against fragmentation. In order to push these disks
into the fragmenting regime, one must invoke highly variable accretion or lower stellar illumination.
Highly variable accretion can force disks like case study (2) up to the fragmentation threshold,
where binary formation is the most likely outcome. Reduced irradiation is more relevant for older
disks, where fragmentation into low mass objects is most likely as we discuss in the following
section. These studies also suggest that burst-like behavior is probable when GI is coupled with
other modes of angular momentum transport like the MRI. Finally, Kratter et al. (2008), who
explored the evolution of higher mass stars, found that more massive stars often evolve from the
self-luminous global mode regime at early times into the irradiated, massive, and fragmenting regime
at late times. The primary difference is that more massive stars are thought to experience higher
mass infall rates, which more easily exceed the maximum GI accretion rate discussed in Section
3.6.
To summarize, we have shown that protostellar and protoplanetary disks sample several of the
regimes discussed in the previous section. Compact, Class 0 disks (see, for example Rodrı́guez
et al. 2005), can be GI-unstable, self-luminous, and regulated via thermal saturation and modemode coupling. For most other cases, a disk that reaches the threshold for gravitational instability
will have a temperature controlled primarily by stellar irradiation, with short cooling times. Under
these circumstances, the disk will either fragment, or the instability will saturate through modemode coupling, depending on whether or not the infall rate exceeds that which can be processed by
strong global modes. As a result, it is imperative to consider the impact of both stellar irradiation
and infall in characterizing the behavior of these disks.
www.annualreviews.org • GI DISKS
27
5. GI and planet formation
The non-linear phase of GI can lead to fragmentation of the disk into bound objects in orbit
about the primary star. What are these objects? Beginning with Adams et al. (1989), there is
a growing consensus that GI-borne objects are most likely to evolve into either brown-dwarfs or
stellar binary companions around sufficiently massive stars (Kratter & Matzner 2006; Kratter et al.
2010b; Stamatellos & Whitworth 2009b; Zhu et al. 2012; Forgan & Rice 2013). In contrast, Boss
(1997) argued that the direct collapse of clumps from gravitational instabilities can create massive
gas giant planets (see also Durisen et al. 2007 and references therein). More recently, a series of
papers starting with Nayakshin (2010), have proposed that a combination of GI and tidal stripping
can even produce rocky planets. Considering which disks are susceptible to GI and fragmentation
reveals parameters unfavorable to the formation of planets, but favorable to the formation of more
massive wide-orbit companions. In Section 4 we have seen that unstable disks are massive, which
leads to large initial masses for fragments, often undergoing rapid infall, which can enhance the
subsequent growth rates, and experiencing rapid angular momentum transport, which can cause
rapid migration and destruction of marginally bound objects (which are the most likely to be low
in mass in the first place). All of these features favor the formation of objects above the deuterium
burning limit.
The disks best suited to forming lower mass objects are those that are unstable in the absence
of continuing infall. These disks tend be less massive, and are thus colder when Q = 1, which
translates to smaller initial masses, less continued growth, and the possibility of slower migration in
a more laminar disk. While these conditions are much more favorable for forming low mass objects,
the formation of close-in companions < 20au, with low mass < 1MJup , is the exception.
5.1. Initial clump conditions
The mass at which a GI perturbation becomes a bound fragment is the outcome of the non-linear
phase of the instability, and is thus challenging to predict from simple analytic models. Although
numerical simulations are now in principle well enough resolved to answer these questions, the
masses evolve from this initial state quickly, and simulations become poorly resolved on a similar
timescale. It is thus valuable to have analytic estimates with which to compare numerical models.
The simplest approximation is that:
Mf ≈
Σ
π
Σλ2 = 4MJup
4
50gcm−2
H/r
0.2
2 R
50AU
−2
(49)
where
λ = 2πH
(50)
is the most unstable wavelength for the axisymmetric instability when Q = 1. The factor of 4
comes from assuming that only 1/2 of the unstable wavelength generates the over density. As
noted by Kratter & Murray-Clay (2011), such a fragment would by definition be unbound at the
Σ associated with Q = 1, because the size implied by λ is larger than the Hill radius for the mass
Mf in Equation (49). This is consistent with the instability producing non-linear perturbations.
Many authors have argued for different, order unity factors in front of the dimensionally motivated Σλ2 estimate (e.g. Boley et al. 2010; Forgan & Rice 2011). In Figure 5.1 we illustrate several
different fragment mass estimates, which span an order of magnitude in mass. To compare these
mass estimates we assume a temperature profile:
T = 30K
28
Kratter & Lodato
r −3/7
70au
(51)
What is a planet?
To determine whether gravitational instability makes planets, one requires a working definition of a planet.
At present, there is no consensus in the community. In this chapter, we use nuclear burning as the dividing
line, although fragmentation is ignorant of this boundary. We also suggest that system architecture be
considered in separating planetary systems from stellar multiples. While planets typical orbit a dominant
central mass (or occassionally a tight binary), higher order stellar systems typically have hierarchical orbits.
which is (conservatively) a ∼ 20% reduction from the fiducial irradiation model of Kratter et al.
(2010b). We set the disk surface density such that Q = 1 at every radius shown. We choose a
critical value of β = 13, (relevant only for the prediction from Forgan & Rice 2011). In addition
to the initial fragment mass estimates, we also show several limiting cases. We plot the canonical
gap opening mass (using α set by the cooling time) and the isolation mass (Lissauer 1987). These
scales are described in more detail in Section 5.1.3. Even the lowest mass estimate predicts initial
fragments above 1MJup for typical disk properties. The estimates of the mass at which fragments
might cease to grow are all well above the deuterium burning limit. Clearly fragments must be
prevented from growing if they are to become planets.
5.1.1. Clump Survival. Regardless of the mass at which clumps are born, they face a variety of
threats to long term survival. These can be categorized as:
1. Dispersal due to interaction with another over density (clump or spiral arm)
2. Tidal shredding due to slow cooling
3. Tidal shredding due to rapid migration
All three of these rely on the balance between contraction due to cooling, which allows the fragment’s
gravity to become ever more dominant, and disruption due to tidal forces from either the star or
another overdensity in the disk.
First consider the cooling rate of a newborn fragment. In Section 3 we discussed the critical
role of the cooling time in determining whether an unstable disk can thermally self-regulate before
the instability becomes non-linear. In modelling clump survival, we must consider the cooling
properties of the clump itself as it separates from the disk and increases in density. Although the
numerical values of the critical cooling time are not wildly different between disk and fragment,
they can in principle be different if there are sharp changes in the equation of state with density.
Moreover, the appropriate radiative cooling timescale for a clump is, in fact, different from that
which governs thermal self-regulation. The former is a comparison of the disk’s thermal energy to
the radiative cooling rate, assuming a vertically isothermal disk. The cooling timescale relevant for
clump survival describes the evolution of a midplane perturbation. The appendix of Kratter et al.
(2010b) works out this timescale as:
tcool

−1
 (1 − B/4) , T >> T0
3γΣc2s τ
×
=
32(γ − 1)σT 4 
1, T h T0
(52)
where T0 is the temperature set by irradiation alone and κ ∝ T B is the dependence of the opacity
on temperature, typically 0 < B < 2. Comparison with Equation (36) shows that the perturbation
cooling timescale is only different by a factor of a few at most. The cooling rate sets the size of
www.annualreviews.org • GI DISKS
29
104
BROey 2010
)Rrgan 2011
Gap 2pening
IsROatiRn 0ass
0ass (MJ )
103
102
101
100
20
40
60
80
100
RaGius (au)
120
140
160
Figure 3
Here we illustrate the range of initial fragment masses with lower and upper bounds set by Boley et al.
(2010) and Forgan & Rice (2011), respectively. We also show estimates for the masses at which accretion
onto the fragment from the background disk might cease: the gap opening mass and the isolation mass.
See text for details about the disk model used for this calculation.
a fragment as a function of time. For typical fragmenting disks, which have β . 10, cooling is
sufficiently fast that fragments initially contract on their free fall timescale (neglecting tidal forces).
The contracting fragment’s radius must be compared with the scale on which tidal forces can
shred it. Naively, one requires that:
rclump < RH =
Mclump
3M∗
1/3
a
(53)
where a is the current semi-major axis. Consider the origin of this limit: the Hill radius RH
determines where an object’s self-gravity will dominate over tidal gravity. In the case of a collapsing
fragment, pressure forces may not be negligible, weakening the inward pull of self-gravity. Kratter &
Murray-Clay (2011) find that tidal shredding can occur at radii of order RHill /3 if pressure support is
important. However, even if fragments suddenly slow their cooling, and undergo Kelvin-Helmholtz
contraction rather than free fall collapse, most will contract to < RHill /3 within a dynamical time,
almost guaranteeing survival at a fixed radius.
30
Kratter & Lodato
Clump interaction. Fragments may also be disrupted due to tidal interaction with other nearby
fragments or spiral arms. Shlosman & Begelman (1989) define a critical cooling time for clumps
to collapse prior to interacting. This is relevant if the disk breaks up into multiple fragments with
separations of order
ptheir own size. Since the encounter velocity of two fragments is roughly the
Hill velocity vH = GMclump /RH , fragments cross each others Hill radius in one dynamical time,
implying that fragments must collapse to well within the Hill radius on this very rapid timescale.
This stringent requirement of collapse in one orbital period is unlikely to be necessary in fragmenting
protostellar disks, which have H/r sufficiently large that fragments form relatively far apart (Boley
et al. 2010; Kratter et al. 2010a, though see Stamatellos & Whitworth 2009a).
In addition to interacting with other clumps, clumps may also interact with a passing spiral
wave, since we know these should be present when fragmentation begins. Young & Clarke (2016)
argued that the quasi-regular nature of the spiral patterns in self-gravitating discs (as first characterized by Cossins et al. 2009) should encourage proto-fragment disruption via collisions with spiral
features over many orbital timescales. Their quantification of the intermittency of spiral features
in simulations suggested that ‘stochastic’ fragmentation should not be possible for β > a few ×10.
5.1.2. The physics of clump cooling. We have outlined above some disruption mechanisms and
the critical dimensionless cooling rates. But what determines the cooling rate physically? The
cooling physics is nearly identical to that at the onset of star formation (Larson 1969), though
self-gravitating clumps will generally be optically thick from the start (Rafikov 2005). At low
temperatures, as in the background disk, cooling rates are set primarily by dust opacity. If grain
growth has already commenced in such disks the opacity might be somewhat reduced compared
to the ISM case. As the clump contracts under its own self-gravity, compressional heating causes
the central temperatures and densities to rise. At temperatures above roughly 1200K (Bell & Lin
1994; Semenov et al. 2003), dust sublimation removes the dominant source of opacity, accelerating
cooling somewhat. Once temperatures rise all the way to 2000K, triggering molecular hydrogen
dissociation, rapid collapse ensues toward planetary sizes.
Galvagni et al. (2012) has carried out a careful study of the cooling timescale of clumps including
a complex equation of state. They extract a clump from Boley et al. (2010) (neglecting subsequent
growth, which can affect both the mass and cooling on similar timescales). They include radiative
cooling due to a mixture of atomic and molecular hydrogen, with an opacity appropriate for ISM
grains. They find that clumps that do not have their rotation artificially suppressed collapse to
roughly 40% of their initial radius within 10 dynamical times. The same clump with rotation
suppressed collapses even faster to roughly 15% of the initial radius over 10 dynamical times. This
suggests that the clumps described by these models would satisfy the constraints given above,
predicting survival at fixed radius.
5.1.3. Continued Growth. A detailed treatment of the evolution of clumps based on their initial
mass provides a useful benchmark, but omits the influence of continued accretion from the background disk. The absolute upper limit is set by the isolation mass, which is the mass that a
secondary potential embedded in the disk can attain by accreting disk material within its own Hill
sphere (Lissauer 1987). The importance of continued accretion following fragment formation is
likely a strong function of initial mass. If the initial mass is very large, it will clear a deep gap
in the surrounding disk, possibly reducing the accretion rate. In contrast, low mass fragments, or
massive fragments in rapidly accreting disks, would remain semi-embedded in the natal disk.
Gaps are opened in a disk when a relatively massive companion produces gravitational torques
which repel material faster than viscosity can replenish it. Using the standard gap opening requirewww.annualreviews.org • GI DISKS
31
ments of Lin & Papaloizou (1986) and Bryden et al. (1999), gaps are opened for mass ratios:
5/2 s
α 1/2 T 5/4 r 5/4 M −5/4
Mclump
H
3πα
∗
(54)
>
≈ 4 × 10−3
M∗
r
fg
0.1
20K
70AU
M
where fg = 0.23 is a geometric factor derived by Lin & Papaloizou (1993). By comparing Equation (54), the red line, with the initial mass estimates shown in Figure 5.1, we see that typical
fragments will be near, but below the gap opening mass, and likely able to grow at least this much.
We benchmark our gap-opening estimate with a relatively large value of α as this is typical for
strongly self-gravitating disks.
Gap-opening, which occurs at masses well below the isolation mass, may not entirely starve
growth. Numerous simulations of Type II planet migration and gap opening show that at least
some material does flow across the gap, and enter into circumplanetary disks. Indeed Duffell et al.
(2014) find that the standard Type II migration rates that assume no mass flow are incorrect for
the same reason. Lissauer et al. (2009) found in numerical simulations that gaps roughly 5RH in
width are capable of entirely shutting off accretion. Whether this limit, obtained in much lower
viscosity disks, applies, is uncertain.
5.1.4. Migration of clumps. Even if clumps survive at fixed radius, as discussed above, they are
subject to tidal torqueing by the parent disk. Fragments may initially be just below the gap
opening mass, suggesting that they begin migrating similarly to planets undergoing type I migration.
Baruteau et al. (2011) have confirmed in 2D simulations that the migration of such objects in
gravitationally unstable disks is quite rapid, and that the gap opening criteria is accurate even in
such disks. They find that despite the presence of stochastic outward forcing due to GI, the overall
migration direction is inwards. Similarly, Zhu et al. (2012) find that planets suffer divergent fates
depending on whether or not they open a gap; gap-opening fragments first move in, then stall their
migration, while smaller planets migrate inwards on < 10 dynamical times. This is consistent with
the work of Vorobyov (2013) in terms of migration timescale. The scenarios in which migration is
most likely to stall or even reverse direction are when fragments become massive compared to the
disk out of which they formed. While very interesting for understanding the distribution of brown
dwarf and stellar binary companions (Kratter et al. 2010a), these are not relevant to the question
of giant planet formation.
5.1.5. Long timescale disruption: Tidal downsizing. As noted at the start of this section, a second
GI planet formation paradigm has arisen in the last few years wherein an initially massive fragment
becomes enriched in solids and migrates inwards until its Hill Radius shrinks below the fragment
radius. The envelope of the fragment is completely or partially stripped, leaving behind a low
mass, rocky core with or without an atmosphere (Nayakshin 2010; Boley et al. 2011). Successive
refinements of this model include the effect of radiative feedback (Nayakshin & Cha 2013), heavy
element sedimentation (Nayakshin et al. 2014), and late stage metal enrichment (Nayakshin 2015)3 .
One significant concern with these models is the attempt to disentangle the initial disk conditions
from the clump evolution. While very sensible from a modeling perspective, this can have an outsize
impact on the conclusions. For example Nayakshin & Cha (2013) use artificially inserted fragments
in a non-fragmenting disk as the initial conditions, which does not adequately capture the resultant
disk-fragment interactions. Similarly, Nayakshin et al. (2014) neglect accretion from the disk onto
3 In Nayakshin (2015), the neglect of heating due to the addition of solids is likely responsible for their
counterintuitive finding that planetesimal accretion accelerates core collapse, which contradicts most previous work in the area (Pollack et al. 1996; Rafikov 2011)
32
Kratter & Lodato
the fragment, which might fundamentally change the conclusions. These issues aside, we show in
the following section that populations synthesis models of this process produce objects which are
inconsistent with the rocky planet population known thus far.
5.2. Comparison with Observed Population
What, if any, insight can be gained from looking at the known population of planets and low
mass companions? The GI model makes some predictions for the populations. The two simplest
predictions come directly from the required initial conditions for clump formation: more massive
stars seem more prone to developing gravitational instabilities and when clumps survive, they are
prone to growing beyond the deuterium burning limit. Forgan & Rice (2013) have carried out
the most extensive population synthesis calculations to date, including a range of initial masses,
accretion of solids, migration, and tidal disruption. Notably, this work excludes any gas accretion
after the initial clump is formed, meaning that their final masses are a lower limit (and much of
the evolution might change if mass growth occurred early). They find that roughly half of the
initially formed clumps are totally destroyed, and 90% of the remaining objects are above the
Deuterium burning limit for a wide variety of initial models. A strong prediction of these models
is that GI mostly makes BDs or more massive objects when it occurs, and even if tidal truncation
/ downsizing can in principal produce some rocky cores, it is a finely tuned part of the parameter
space and unlikely to be responsible for the bulk of observed rocky planets.
In Figure 4, we illustrate the population of low mass companions as compiled by the exoplanets.eu catalog. This catalog includes companions up to 30MJup . We show companions as a function
of mass ratio, rather than measured mass, as this has more relevance to the question of GI. Note
that most of the Kepler planets are excluded from this diagram as their masses are unknown. Based
on their periods and radii, they primarily occupy the region within 1AU of the host star and below
10−5 in mass ratio. GI is most likely to produce the objects at 10s of au above mass ratios of
10−2.5 . In order for GI to account for more of the “normal” planet population this outer region
would likely be more populated than allowed for by current observational limits (Forgan & Rice
2013).
On the opposite end of the mass spectrum, is their observational evidence for GI as a mechanism
for BD and binary formation? Yes and no. On the one hand, the propensity of massive stars to
binary formation via disk fragmentation is generally consistent with the abundance of binaries
around more massive stars (Raghavan et al. 2010; Sana & Evans 2010). On the contrary, Kraus
et al. (2011) find that for young stars in Taurus, the lower mass objects M∗ < 0.7M typically
have companions with smaller separations consistent with disk fragmentation (< 100AU ,) whereas
stars with 0.7M < M∗ < 2.5M have more companions at larger separations. Recent work on
Class 0 and I binaries now finds evidence for a bimodal distribution of separations, characteristic
of a disk and core mode of fragmentation (Tobin et al. 2016). Dupuy & Liu (2011) found that the
orbital distribution of BD binaries was inconsistent with ejection in pairs from fragmenting disks
as predicted by Stamatellos & Whitworth (2009a). As of this writing there is not a complete model
for binary formation from cores and disks. Bate (2012) provides the largest statistical sample of
simulated binary formation, which matches well with observations of the field. Unfortunately the
disk fragmentation process is not sufficiently well resolved in these calculations to attribute parts
of the binary distribution to one mechanism.
5.3. Role of GI in the core accretion model for planet formation
GI may have an entirely separate means of kickstarting the planet formation process: by concentrating grains in spiral arms. Protoplanetary disks can be well-modelled as two fluids: gas and
www.annualreviews.org • GI DISKS
33
101
100
10-1
10-2
Ms =M ¤
10-3
10-4
10-5
10-6
All Objects
Ms >13MJ
M ¤ >1:5M ¯
M ¤ <0:5M ¯
10-7
10-8 -3
10
10-2
10-1
100
101
102
Semi-Major Axis (AU)
103
104
Figure 4
The distribution of planetary and other low mass companions from Exoplanets.eu as a function of mass
ratio versus semi-major axis. All objects are shown in green, those above the deuterium burning limit are
demarcated by red dots, those with more massive primaries are shown in yellow, while those with M-star
hosts are green
solids. As noted in Section 3, while the gas feels pressure forces, which counteract gravity and
slow orbital velocities, solids act as a pressureless fluid. The relative velocities between the two
fluids generates drag, which is a function of particle size. This drag force results in slow radial drift
of particles towards pressure maxima (Whipple 1972; Adachi et al. 1976; Weidenschilling 1977).
We neglect in this review the separate physics of self-gravity of solid layer in protoplanetary disks
(Goldreich & Ward 1973; Weidenschilling 1980), or self-gravity of planetesimals concentrated by
the streaming instability (Youdin & Goodman 2005).
Spiral arms in a self-gravitating disk are an excellent candidate for producing pressure maxima
that can enhance particle growth, much like MRI turbulence for the Streaming Instability (Johansen
et al. 2009). Rice et al. (2004) using 3D, global SPH simulations of gas and dust coupled via drag
with a simple algorithm first demonstrated numerically that these traps work effectively in selfgravitating disks. Later, Rice et al. (2006), including the dust self-gravity, showed that the dust
sub-structure induced by the traps can become self-gravitating itself, and lead to a rapid formation
of large mass rocky cores. The actual mass of these cores is related to the Jeans mass of the
dust substructure, which in turn depends on the velocity dispersion induced by the gas flow in the
particle distribution. Whether the velocity dispersion measured in such simulations (typically of
34
Kratter & Lodato
the order of the sound speed) is physical or affected by numerics is still to be fully understood (see
Booth et al. 2015).
Clarke & Lodato (2009) have raised the concern that this effect might require the cooling
timescale in the disc to be relatively small based on the magnitude of the radial velocity induced by
an overdensity of a given size. This would confine the effect to the outer regions of the disk. However,
Gibbons et al. (2012), using 2D grid based simulations conclude that significant clumping occurs
even for tcool ≈ 40Ω−1 , much above the fragmentation boundary. The particles with stopping times
τs of order unity will typically be 1-10cm in size for the densities associated with self-gravitating
disks at radii of 10-100AU. Gibbons et al. (2014) also consider the self-gravity of the particles and
their back reaction on the gas. Again, similar to the Streaming Instability (Youdin & Goodman
2005), they find that particle concentrations in spiral arms can become high enough to become
self-gravitating and collapse directly into large planetesimals.
A global view of particle clumping due to GI with an appropriate size distribution for the dust
would be required to fully characterize the effect that GI might have in the core accretion model.
This requires the use of an accurate and versatile algorithm for simulating the coupled dust and
gas dynamics (Laibe & Price 2012, 2014; Price & Laibe 2015).
Another issue to be fully understood is the fate of the rocky cores or planetesimals formed
in a self-gravitating disk as described above. Generally, the mass of the rocky objects formed by
particle concentrations within spiral arms is not expected to be high enough to directly trigger
core accretion and significant further growth is required. However, such growth is significantly
hampered during the self-gravitating disk phase, where the gas structure (even in the absence of
drag) induces gravitational potential fluctuations that pump the eccentricity of the planetesimal
swarm up to very high values, thus inhibiting significant growth (Britsch et al. 2008). Such an early
population of planetesimals can thus produce large rocky cores only if they are retained in the disk
long enough that the disk has become GI stable (Walmswell et al. 2013).
6. Conclusions and Outlook
In this review, we have provided a broad overview of the physical nature of GI, illustrating the
variety of ways in which the instability manifests itself. It is first and foremost an angular momentum transport mechanism, and under some circumstances it may cause disk fragmentation,
which can lead to the formation of secondary companions. We demonstrate that protostellar disks
around low mass stars may be prone to a self-regulated form of the instability at early times, if their
mass ratios are high. At later times and larger radii, disk are irradiation dominated and prone to
fragmentation if unstable. The resultant fragments are unlikely to be below the deuterium burning
limit. Protoplanetary disks are not expected to be unstable based on current observations. Disks
around more massive stars are more likely prone to both instability and fragmentation because
rapid infall drives the disks to higher masses.
There are a handful of theoretical issues that remain unresolved. The first set of open questions
relate to saturation via mode coupling. Can mode coupling operate in tandem with thermal regulation? Under what circumstance does mode-mode coupling dominate over thermal regulation?
Is there a critical disk-star mass ratio at which a transition occurs? How is the maximum mode
amplitude set? Does mode coupling change under rapid infall? Since this form of saturation may
dominate in protostellar disks, a detailed understanding of the mechanism is warranted. The second
set of open questions relates to the critical cooling time, the nature of so-called gravito-turbulence,
and stochastic fragmentation. Does GI produce a turbulent energy cascade and power spectrum
akin to isotropic turbulence in the ISM? Or is gravito-turbulence observed in simulations the high
resolution manifestation of small-m spiral mode growth? If GI does lead to turbulence, then the
www.annualreviews.org • GI DISKS
35
stochastic fragmentation at a range of cooling times is viable. If, on the other hand, the instability
does not produce rare, high amplitude density perturbations, there should be a concrete value of
the cooling time at which self-regulated disks transition to fragmentation
On the observational front, much progress has been made in recent years, and ALMA is poised
to make significant breakthroughs as all dishes come online. Protostellar disks are now being imaged
with high enough resolution to reveal their internal structure and morphology with unprecedented
details. As mentioned above, in several cases a prominent spiral structure can be observed (for
example, in the NIR scattered light images of MWC 758, Benisty et al. 2015). Several papers have
anticipated the exceptional ability of ALMA to detect spiral structures induced by GI both through
molecular line emission (Krumholz et al. 2007) and through dust emission (Cossins et al. 2010b;
Dipierro et al. 2014). It will be possible to distinguish the large scale, open spirals expected in
massive disks (Krumholz et al. 2007) from the more tightly wound spirals expected in the low mass
case (Dipierro et al. 2014). ALMA might even be able to detect the spatial changes in opacity due
to dust trapping in spiral arms (Dipierro et al. 2015).
SUMMARY POINTS
1. Gravitational Instability provides efficient angular momentum transport in relatively massive protostellar disks. Evidence of this phase observationally remains sparse, but is likely
to improve in the coming years with more ALMA data.
2. Based on the models presented in this review, we find that protostellar disks which achieve
Q ∼ 1 often have large disk-star mass ratios and temperatures controlled by stellar irradiation. In this regime, Q evolves due to changes in Σ rather than T , and therefore infall is
the dominant driver of instability.
3. When fragmentation does occur, BD or stellar companion formation is more likely than
planet formation. Evidence that the instability ever forms planets is currently lacking. Uncertainties in the fragmentation boundary observed in numerical simulations have minimal
impact on the likelihood of planet formation via this mechanism.
FUTURE ISSUES
1. Demonstration of the relevance of the instability to realistic disks requires high resolution,
high sensitivity observations with ALMA coupled with sophisticated molecular line and
radiative transfer models that can more reliably ascertain disk masses.
2. In protostellar disks, perhaps the main channel by which gravitational instability saturates
– mode-mode coupling – remains poorly understood and should be the subject of future
investigation.
3. Gravito-turbulence has not convincingly been shown to behave like turbulence with a multiscale power spectrum and energy cascade. The extent to which gravito-turbulence exhibits
these characteristics is intimately tied to the issue of a critical cooling time and stochastic
fragmentation. Stochastic fragmentation requires a very well sampled PDF, which may or
may not occur in realistic disks.
36
Kratter & Lodato
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdings that
might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
The authors would like to thank Phil Armitage, Cathie Clarke, and Jordan Stone for comments on
earlier drafts of this article. G.L. was supported by the PRIN MIUR 2010-2011 project “The Chemical and Dynamical Evolution of the Milky Way and Local Group Galaxies”, prot. 2010LY5N2T.
K.M.K. was supported in part by the Transition Grant of the Universita’ degli Studi di Milano.
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